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Abstract
Let R be a commutative non-domain ring with identity and let denote the set of all nonzero annihilating ideals of R. Recall that the annihilating-ideal graph of R, denoted by
, is an undirected simple graph whose vertex set is
and distinct vertices I, J are joined by an edge in this graph if and only if
. The aim of this article was to classify commutative rings R such that the independence number of
is less than or equal to four.
Public Interest Statement
The research work carried out by I. Beck in the year 1988 inspired a lot of researchers to investigate the interplay between graph theory and ring theory. The basic purpose of this research was to study the structure of commutative rings with the help of annihilating-ideal graphs of rings. Indeed, in this article, we focused our study on the independence number of annihilating-ideal graphs of commutative rings. The outcome of this study is that we are able to classify all commutative rings whose annihilating-ideal graph has independence number at most four.
1. Introduction
The rings considered in this article are commutative with identity and which are not integral domains. The concept of associating a ring with a graph and investigating the interplay between the ring theoretic properties of rings under consideration and the graph theoretic properties of the graphs associated with them was initiated by Beck (Citation1988). In Beck (Citation1988), I. Beck was mainly interested in colorings. The work of I. Beck inspired a lot of research activity in the area of associating graphs with algebraic structures and exploring the influence of certain graph theoretic parameters on the algebraic structure of the considered algebraic objects. Let R be a ring. The concept of zero-divisor graph of R, denoted by , was introduced and studied by Anderson and Livingston (Citation1999). Recall from Anderson and Livingston (Citation1999) that
is an undirected simple graph, whose vertex set is
, where Z(R) is the set of all zero-divisors of R and
and two distinct vertices
are joined by an edge in this graph if and only if
. During the last two decades several mathematicians have contributed to the area of zero-divisor graphs in commutative ring, to mention a few, refer Anderson, Frazier, Lauve, and Livingston (Citation2001), Anderson and Livingston (Citation1999), Anderson and Naseer (Citation1993), Axtell, Coykendall and Stickles (Citation2005), Lucas (Citation2006), Smith (Citation2003). For an excellent and clear exposition of the work done in the area of zero-divisor graphs in commutative rings, the reader is referred to the following survey article Anderson, Axtell and Stickles (Citation2011).
Let R be a ring. Recall from Behboodi and Rakeei (Citation2011a) that an ideal I of R is said to be an annihilating ideal if for some
. The concept of the annihilating-ideal graph of R, denoted by
, was introduced in Behboodi and Rakeei (Citation2011a) by M. Behboodi and Z. Rakeei. Recall form Behboodi and Rakeei (Citation2011a) that
is an undirected simple graph whose vertex set is
, where
is the set of all annihilating ideals of R and
, and two distinct vertices I, J are joined by an edge in this graph if and only if
. The interplay between the ring theoretic properties of R and the graph theoretic properties of
was very well investigated in Behboodi and Rakeei (Citation2011a,Citation2011b) and several interesting theorems were proved in Behboodi and Rakeei (Citation2011a,Citation2011b) on
indicating the effect of certain graph theoretic parameters on the structure of R. Moreover, the annihilating-ideal graph of a commutative ring was also studied by several others, to mention a few, the reader is referred to Aalipour et al. (Citation2014), Aalipour, Akbari, Nikandish, Nikmehr, and Shaiveisi (Citation2012), Hadian (Citation2012).
The graphs considered in this article are undirected. Let be a simple graph. We denote the complement of a graph G by
and we denote the complete graph on n vertices by
. Recall from Balakrishnan and Ranganathan (Citation2000, Definition 1.1.4) that a vertex u is a neighbor of v in G, if there is an edge of G joining u and v. A clique of G is a complete subgraph of G (Balakrishnan & Ranganathan, Citation2000, Definition 1.2.2). If the size of the cliques of a graph G is bounded, then the clique number of G, denoted by
, is the largest positive integer n such that G contains a clique on n vertices (Balakrishnan & Ranganathan, Citation2000, p. 185). If G contains a clique on n vertices for all
, then we set
.
Let be a graph. A vertex coloring of G is a map
, where S is a set of distinct colors and a vertex coloring f is said to be proper, if adjacent vertices of G receive distinct colors of S: that is, if u, v are adjacent in G, then
(Balakrishnan & Ranganathan, Citation2000, p. 129). Recall from Balakrishnan and Ranganathan (Citation2000, Definition 7.1.3) that the chromatic number of G, denoted by
, is the minimum number of colors needed for a proper vertex coloring of G. It is well known that
.
Let be a graph. Recall from Balakrishnan and Ranganathan (Citation2000, Definition 5.1.1) that a subset S of V is called independent if no two vertices of G are adjacent in G.
is a maximum independent set of G if G has no independent set
with
. The number of vertices in a maximum independent set of G is called the independence number of G and is denoted by
(Balakrishnan & Ranganathan, Citation2000, Definition 5.1.4). It is well known that for any simple graph G,
(Balakrishnan & Ranganathan, Citation2000, p. 186).
The clique number and chromatic number of zero-divisor graphs of commutative rings have been studied by several researchers Anderson et al. (Citation2001), Anderson and Naseer (Citation1993), Beck (Citation1988), Smith (Citation2003). A good account of the work done on the clique number and chromatic number of the zero-divisor graphs of commutative rings has been given in Anderson et al. (Citation2011). Let . Rings R such that
as determined by Beck (Citation1988), and by Anderson and Naseer (Citation1993) were listed in (Citation2011). Moreover, Smith (Citation2003) has classified all finite commutative nonlocal rings R such that
. Section 3 of Visweswaran (Citation2011) contains some results on the clique number of
. The study of the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring was carried out in Aalipour et al. (Citation2012), Behboodi and Rakeei (Citation2011b). Let
. Inspired by the above-mentioned works, in this article, we try to classify commutative rings R such that
. We are also interested to determine the least integer
with
. Observe that
. As is suggested by the referee, we focus our study on classifying rings R such that
.
It is useful to recall the following results from commutative ring theory that we use in this article. Let I be an ideal of a ring R with . A prime ideal
of R is said to be a maximal N-prime of I, if
is maximal with respect to the property of being contained in
for some
(Heinzer & Ohm, Citation1972). Thus a prime ideal
of R is a maximal N-prime of (0) if
is maximal with respect to the property of being contained in Z(R). Observe that
is a multiplicatively closed subset of R. Let
. Then
. Hence, it follows from Zorn’s lemma and (Kaplansky, Citation1974, Theorem 1) that there exists a maximal N-prime
of (0) in R such that
. Therefore, we obtain that
, where
is the set of all maximal N-primes of (0) in R.
Recall that a principal ideal ring is a special principal ideal ring (SPIR) if it has a unique prime ideal. If R is a SPIR with as its only prime ideal, then we denote it using the notation
is a SPIR.
We say that a graph G satisfies (C) if G does not contain any infinite clique. Let R be a ring. In this article, we often use some of the results that were proved in Visweswaran and Patel (Citation2015) on rings R such that satisfies (C). Let R be a ring with at least two maximal N-primes of (0). It is proved in Visweswaran and Patel (Citation2015, Theorem 3.1) that
satisfies (C) if and only if
as rings for some
, where
is a local ring which admits only a finite number of ideals for each
, if and only if
. Moreover, for a ring R with exactly one maximal N-prime of (0), it is not known whether the conditions that
satisfies (C) and
are equivalent. Motivated by Visweswaran and Patel (Citation2015, Theorem 3.1), we propose to find the precise characterization of rings R, at least for smaller values of
. In Section 2 of this article, we classify rings R such that
, where
. In Section 3, we focus our study on classifying rings R such that
. We devote Section 4 of this article on the problem of classifying rings R such that
.
We denote the nilradical of a ring R by nil(R). A ring R with a unique maximal ideal is referred to as quasilocal and a ring R with only a finite number of maximal ideals is referred to as semiquasilocal. A Noetherian quasilocal (respectively, a semiquasilocal) ring R is referred to as local (respectively, semilocal). We use |A| to denote the cardinality of a set A.
2. Classification of rings R such that ![](//:0)
![](//:0)
We start this section with the following lemma.
Lemma 2.1
Let R be a ring which admits at least n maximal N-primes of (0) with . Then
.
Proof
The conclusion of the lemma holds if contains an infinite independent set. Hence, we can assume that
does not admit any infinite independent set. In such a case, we know from Visweswaran and Patel (Citation2015, Theorem 3.1) (the statement of this theorem is already stated in the introduction) that R is necessarily Artinian. Therefore, each proper ideal of R is an annihilating ideal. Let
be a subset of the set of all maximal ideals R. Let
. Let
. Since any two distinct maximal ideals R are not comparable under the inclusion relation, it follows from Prime avoidance lemma (Atiyah & Macdonald, Citation1969, Proposition 1.11(i)) that
. Hence, for each
, there exists
but
. It is clear that
for all distinct
. Let
,
. Since
, there exists
. Observe that
and hence,
. It is evident from the above discussion that
is an independent set of
. This proves that
.
In this section, our interest is to classify rings R such that . In view of Visweswaran and Patel (Citation2015, Theorem 3.1), we assume in the hypothesis of many of the auxillary results we prove in this section that R is a finite direct product of rings.
Lemma 2.2
Let and let
be a quasilocal ring for each
such that
is an annihilating ideal for each i. Let
be their direct product. Then the following hold:
(i) | Let | ||||
(ii) | If | ||||
(iii) | If | ||||
(iv) | Let |
Proof
(i) | Without loss of generality, we can assume that | ||||
(ii) | Without loss of generality, we can assume that | ||||
(iii) | Without loss of generality, we can assume that | ||||
(iv) | We are assuming that |
Lemma 2.3
Let and let
be a ring for each
. Let
be their direct product. Then
.
Proof
Let , and
. Observe that
for each
and
is an independent set of
. This proves that
.
We next proceed to characterize rings R such that . It follows from Lemma 2.1 that such rings R can admit at most two maximal N-primes of (0). In Propositions 2.4 and 2.6, we classify rings R such that
. In view of Visweswaran and Patel (Citation2015, Theorem 3.1), whenever a ring R admits at least two maximal N-primes of (0) with
, we assume that R is an Artinian ring.
Proposition 2.4
Let R be an Artinian ring which admits exactly two maximal ideals. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Let
denote the set of all maximal ideals of R. Then
. Hence, there exist
and
such that
. It is clear that
and
. Let
. Since R is Artinian, any proper ideal of R is an annihilating ideal. Observe that
,
and
. It follows from
that
. Since
, we get that
. This shows that
. Hence, it follows from the Chinese remainder theorem (Atiyah & Macdonald, Citation1969, Proposition 1.10(ii) and (iii)) that the mapping
defined by
is an isomorphism of rings. Thus with
for each
, we obtain that
,
are fields and
as rings.
This is obvious.
Remark 2.5
Let R be a ring. Let be such that
. If
, then
.
Proof
This is obvious.
We state Visweswaran and Patel (Citation2015, Remark 3.5(ii)) here as we use it in the proof of Proposition 2.6. Let R be a ring which admits as its unique maximal N-prime of (0). If
satisfies (C) and
for some
, then R is necessarily a local Artinian ring with
as its unique maximal ideal.
Proposition 2.6
Let R be a ring which admits as its unique maximal N-prime of (0). Then the following statements are equivalent:
(i) |
| ||||
(ii) | Either |
Proof
Observe that
. Suppose that
. Then there exist
such that
. Hence,
. As
, it follows that
. Therefore, from Remark 2.5, we obtain that
and
. It follows from Visweswaran and Patel (Citation2015, Remark 3.5(ii)) that R is a local Artinian ring with
as its unique maximal ideal. Hence,
. From
, we obtain that
. As
, it follows that
. Hence, it follows from
of (Atiyah & Macdonald, Citation1969, Proposition 8.8) that
is the set of all nonzero proper ideals of R. Therefore,
is a SPIR with
but
.
If
, then
for all
. Otherwise,
is a SPIR with
as its set of all nonzero proper ideals and
. Therefore,
.
Proposition 2.7
Let , where
is a SPIR and
is a field. Let
be the least integer with
. Then
.
Proof
Let . Then
. It is clear that
is the path
. Therefore,
. So, we can assume that
. We consider two cases.
Case(i). is even
Note that . Let
=
,
=
. It is clear that
=
. Let
and let
. Let
and let
. Observe that
is an independent set of
. Since W contains exactly
elements, it follows that
. We next show that there exists a proper vertex coloring of
that makes use of n colors. Let
be a set of n distinct colors. We now color the vertices of
as follows: Let us assign the color
to
for each
. Let the color
be assigned to
for each
and color the vertex
using
. We next assign colors to the vertices of
which are not in W. Let us assign the color
to
for each
. Let us assign the color
to
for each
. Let the color
be assigned to
. It is easy to verify that the above-described assignment of colors using n colors is indeed a proper vertex coloring of
. Thus
. Therefore,
.
Case(ii). is odd
Observe that . Note that
is an independent set of
. Hence,
. Let
be a set of n distinct colors. Let us now assign colors to the vertices of
as follows: Let us assign the color
to
for each
. Let us assign the color
to
for each
. Let the color
be assigned to
. Let the color
be assigned to
for each
. Let the color
be assigned to
for each
. Let the color
be assigned to
. It is easy to show that the above-described vertex coloring of
using n colors is in fact a proper coloring. Thus
. Therefore,
.
Proposition 2.8
Let R be an Artinian ring which admits exactly two maximal ideals. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
It follows from (Atiyah & Macdonald, Citation1969, Theorem 8.7) that
as rings, where
is a local Artinian ring for each
. Since
, it follows that at least one between
and
cannot be a field. Without loss of generality, we can assume that
is not a field. Moreover, we obtain from Lemma 2.2(ii) that
must be a field. Furthermore, it follows from Lemma 2.2(iii) and (Atiyah & Macdonald, Citation1969, Proposition 2.8) that
is principal. Thus
is a SPIR. It is now clear from Proposition 2.7 that
. With
, and
, we obtain that
is a SPIR with
but
,
is a field, and
as rings.
This follows immediately from Proposition 2.7.
For the sake of convenient reference, we state Visweswaran and Patel (Citation2015, Lemmas 2.1 and 2.2), which we use in the proof of Lemma 2.9. Let R be a ring. If satifies (C), then
, where
is the set of all maximal N-primes of (0) in R Visweswaran and Patel (Citation2015, Lemma 2.1). If
satisfies (C), then
Visweswaran and Patel (Citation2015, Lemma 2.2).
Lemma 2.9
Let R be a ring which admits as its unique maximal N-prime of (0). If
, then
is principal.
Proof
We know from Visweswaran and Patel (Citation2015, Lemmas 2.1 and 2.2) that and
. Suppose that
is not principal. Observe that any independent set of
containing exactly two elements must contain
as a member. Let
be an independent set of
. From
, it follows that
. Hence, there exist
such that
. Therefore,
. Since
for each
and
, it follows that
. Therefore, we obtain from Remark 2.5 that
and
. Let
. Since
, it follows that
. Observe that
is an independent set of
. This is in contradiction to the assumption that
. Therefore,
is principal.
Using Visweswaran and Patel (Citation2015, Remark 3.5(ii)), Lemma 2.9, and (Citation2015, Proposition 3.7), it is not hard to prove Proposition 2.10. We state Visweswaran and Patel (Citation2015, Proposition 3.7) for the sake of convenience. Let be a SPIR with
. Let
be the least integer with
. Then
, where [n / 2] is the integral part of n / 2.
Proposition 2.10
Let R be a ring which admits as its unique maximal N-prime of (0). Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
3. Classification of rings R such that ![](//:0)
![](//:0)
The aim of this section was to classify rings R such that . It follows from Lemma 2.1 that such rings R can admit at most three maximal N-primes of (0). We use
of Visweswaran and Patel (Citation2015, Theorem 3.2) in the proof of Proposition 3.1. The statement of
of Visweswaran and Patel (Citation2015, Theorem 3.2) is as follows: Let
and let
be a field for each
. Let
. Then
.
Proposition 3.1
Let R be an Artinian ring which admits exactly three maximal ideals. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
It follows from (Atiyah & Macdonald, Citation1969, Theorem 8.7) that as rings, where
is a local Artinian ring. From
, we obtain from Lemma 2.2(iv) that
is a field for each
. With
for each
, we obtain that
is a field and
as rings.
This is an immediate corollary to
of Visweswaran and Patel (Citation2015, Theorem 3.2).
Proposition 3.2
Let R be an Artinian ring with exactly two maximal ideals. Then the following statements are equivalent:
(i) .
(ii) as rings, where
is a SPIR with
but
and
is a field.
Proof
From (Atiyah & Macdonald, Citation1969, Theorem 3.1), we obtain that
as rings, where
is a local Artinian ring for each
. From
, it follows that at least one between
and
cannot be a field. Without loss of generality, we can assume that
is not a field. Observe that we obtain from Lemma 2.2(ii) that
must be a field. Moreover, it follows from Lemma 2.2(iii) and (Atiyah & Macdonald, Citation1969, Proposition 2.8) that
is principal. Hence,
is a SPIR. Furthermore, we obtain from Proposition 2.7 that
but
. With
, and
, it follows that
as rings, where
is a SPIR with
but
and
is a field.
This follows from Proposition 2.7.
Let R be a ring which admits a unique maximal N-prime of (0). We next proceed to classify such rings R for which . We need several prelimary results to obtain the required classification. We state and prove them in the form of several lemmas.
Lemma 3.3
Let R be a ring which admits as its unique maximal N-prime of (0). If
, then
for some
.
Proof
It follows from Visweswaran and Patel (Citation2015, Lemmas 2.1 and 2.2) that . Let
be an independent set of
. From
, it follows that
. If
is principal, then it is clear that
for some
. So we can assume that
is not principal. From
, it follows that there exist
such that
. We claim that either
or
. Suppose that
. Then it follows from Remark 2.5 that
and
. Hence,
. Note that
is an independent set of
. This is in contradiction to the assumption that
. Therefore, either
or
.
It follows from Lemma 3.3 and Visweswaran and Patel (Citation2015, Remark 3.5(ii)) that if a ring R which admits as its unique maximal N-prime of (0) is such that
, then R is necessarily a local Artinian ring with
as its unique maximal ideal.
Lemma 3.4
Let be a local Artinian ring with
. If
, then
is generated by at most two elements.
Proof
If requires more than two generators, then
.. Let
be such that
is a basis of
as a vector space over
. Then it follows from (Atiyah & Macdonald, Citation1969, Proposition 2.8) that
. From
, it follows that either
for some
or
for some distinct
.
Case(i). for some
Without loss of generality, we can assume that . Note that
is an independent set of
. If either
or
, then
is an independent set of
. This is impossible since by hypothesis,
. Hence,
. Note that
is an independent set of
. This is impossible.
Case(ii). for some distinct
Without loss of generality, we can assume that . The impossibility of Case(i) implies that
. Observe that
is an independent set of
. This is impossible.
This shows that is generated by at most two elements.
Proposition 3.5
Let be a local Artinian ring with
. If
is principal, then the following statements are equivalent:
(i) .
(ii) is a SPIR with either
but
or
but
.
Proof
The proof of this proposition follows immediately from of (Atiyah & Macdonald, Citation1969, Proposition 8.8) and Visweswaran and Patel (Citation2015, Proposition 3.7).
Let be as in the statement of Lemma 3.3. We next assume that
is not principal and try to classify rings R in order that
. Initially, we derive several necessary conditions for
.
Lemma 3.6
Let be a local Artinian ring . If
is not principal and
, then
.
Proof
We know from Lemma 3.4 that there exist such that
. If
and
, then we obtain that
is an independent set of
. This contradicts the assumption that
. Hence, either
or
. Without loss of generality, we can assume that
. We assert that
. This is clear if
. If
, then
is an independent set of
. Since
and
, it follows that either
or
. Therefore, either
or
. From
, it follows that
. Thus
. If
, then
is an independent set of
. This is impossible since by assumption,
. Therefore,
. Similarly, we obtain that
. This proves that
.
Remark 3.7
Let be a local Artinian ring. Suppose that
is not principal. If
, then there exist
such that
with
.
Proof
The proof of this remark is contained in the proof of Lemma 3.6.
Lemma 3.8
Let be a local Artinian ring. Suppose that
is not principal. If
, then
and moreover,
is principal.
Proof
It is proved in Lemma 3.6 that . We know from Remark 3.7 that there exist
such that
and
. Hence,
. Since
, it follows that either
or
. Without loss of generality, we can assume that
. Note that
is an independent set of
. Let
be such that
. Then
. Since
and
, it follows that
. This implies that
. Let
be such that
. Then
. Hence,
. If
, then we obtain
and so from
, it follows that
. This is a contradiction. Therefore,
. This proves that
.
As in the previous paragraph, with
but
. Moreover,
. Now
. If
, then
is principal. Suppose that
. Observe that
is an independent set of
. Note that
and
. Since
, it follows that
. Hence, there exists a unit
such that
. This implies that
. Therefore,
. Let
for some
. Therefore,
. Let
for some
. Then
. Thus
. Since
, it follows that
. Hence,
and so
is principal.
Indeed, in the case , we verify that
. Note that
is an independent set of
. Since
and
, it follows that
. This proves that
.
Lemma 3.9
Let be a local Artinian ring . Suppose that
is not principal. If
, then
.
Proof
Since is not principal, it follows from Lemma 3.4 that
. We know from Lemma 3.6 that
. From Lemma 3.8, we know that
and
is principal. As
is an one-dimensional vector space over
and
, it follows that
. Since
is a two-dimensional vector space over
, it follows that
. Hence,
and therefore,
.
For any prime number p and , we denote by
, the finite field containing exactly
elements . For any
, we denote by
, the ring of integers modulo n.
Remark 3.10
With the help of theorems proved by Corbas and Williams (Citation2000a), Belshoff and Chapman (Citation2007, p. 475) listed (up to isomorphism of rings) all finite commutative rings with identity which are local and of order 16 and there are 21 such rings. In this remark, with the help of the list given in Belshoff and Chapman (Citation2007, p. 475), we list below (up to isomorphism of rings) all finite local rings of order 16 such that
for some
,
, and
..
(i) | |||||
(ii) | |||||
(iii) | |||||
(iv) | |||||
(v) | |||||
(vi) | |||||
(vii) | |||||
(viii) |
|
(a) | |||||
(b) | |||||
(c) | |||||
(d) | |||||
(e) | |||||
(f) | |||||
(g) | |||||
(h) | |||||
(i) | |||||
(j) | |||||
(k) | |||||
(l) |
Lemma 3.11
Let be a local Artinian ring such that
is not principal,
for some
with
but
,
,
, and
. Then
.
Proof
It is clear that and
. Let
. Note that
vary over A
. Observe that
is an independent set of
. Hence,
. We next verify that
.
We first determine the set of all nonzero proper ideals of R. Let I be any nonzero proper ideal of R. If , then it is clear that
. Suppose that
. Then there exists
for some
with at least one between x and y is equal to 1. We consider the following cases.
Case(i).
In this case, it follows from that
. As
, we get that
or 2. If
, then
. If
, then I is principal and indeed
.
Case( ii ).
In this case, . Let
. Then
. Note that R / C is local with
as its unique maximal ideal. As
is principal and
, it follows from
of (Atiyah & Macdonald, Citation1969, Proposition 8.8) that
is the set of all proper ideals of R / C. Therefore, we obtain that
.
Observe that either or
. If
, then we claim that
. For if
, then
for some
. As
for some
, it follows that
. This is a contradiction. Therefore,
. In such a case, it follows from the above discussion that the set of all nonzero proper ideals of R equals
. As
and
, the ideals
are isolated vertices of
. Thus
is the union of the cycle
of length 3 and the isolated vertices. Hence,
. If
, then as
, it follows that
. In this case, it is clear that the set of all nonzero proper ideals of R equals
. Observe that
is the union of the cycles
,
, each of length 3, and the isolated vertex
. Note that
is the edge common to both
and
and Ra, Rb are not adjacent in
. Now it is clear that
.
Thus . Therefore,
.
Example 3.12
With the help of Lemma 3.11, we now verify that each one of the ring R mentioned in (i) to (viii) of Remark 3.10 satisfies .
(I) | Let | ||||
(II) | Let | ||||
(III) | Let | ||||
(IV) | Let | ||||
(V) | Let | ||||
(VI) | Let | ||||
(VII) | Let | ||||
(VIII) | Let |
Let be a finite local ring such that
is not principal and
. Suppose that
. Then we know from Lemma 3.3 that
for some
. We know from Lemma 3.6 that
. Moreover, we know from Remark 3.7 that there exist
such that
with
. As
, we can assume that
. Note that it follows from the proof of Lemma 3.8 that
,
, and
. In Example 3.14, we provide some examples of finite local rings
of order 81 such that
. We use of Lemma 3.13 to verify Example 3.14.
Lemma 3.13
Let be a local Artinian ring such that
is not principal, but there exist
with
and
. If
,
, then
.
Proof
It follows from ,
, and
that
and
. From
, we obtain that
and so
. Moreover, it follows from the given hypotheses that
. Let
. It is then clear that
vary over A
. Let I be any nonzero proper ideal of R. If
, then it is clear that
. Suppose that
. Then there exists an element
with
such that at least one between x and y is different from 0. Then it follows that
and so
. Hence, we obtain that either
or 2. If
, then
. If
, then
. In this case, it is not hard to show that
. This proves that the set of all nonzero proper ideals of R equals
. Note that
is an independent set of
. Therefore,
. We next verify that
. Let
be a set of three distinct colors. Since
, Ra and Rb are not adjacent in
. As
, it follows that
and
are not adjacent in
. From
, it is clear that
is an isolated vertex of
. Observe that
is the union of the cycles
,
, the edges
, and the isolated vertex
. Let us assign the color
to Ra, Rb, and
, color
to
, and the color
to
and
. It is easy to see that the above assignment of colors is indeed a proper coloring of the vertices of
. Hence,
. Therefore,
. This proves that
.
Example 3.14
With the help of Lemma 3.13, we provide some examples of finite local rings with
such that
.
(A) | Let | ||||
(B) | Let | ||||
(C) | Let |
4. Classification of rings R such that ![](//:0)
![](//:0)
In this section we try to classify rings R such that . It follows from Lemma 2.1 that such a ring R can have at most four maximal N-primes of (0). Lemma 4.1 provides the precise number of maximal N-primes of (0) for a ring R with
.
Lemma 4.1
Let R be a ring such that . Then either R has a unique maximal N-prime of (0) or has exactly two maximal N-primes of (0).
Proof
We know from Lemma 2.1 that R can admit at most four maximal N-primes of (0). Suppose to the contrary that R has exactly n maximal N-primes of (0) with . Then we know from
of Visweswaran and Patel (Citation2015, Theorem 3.1) that
as rings with
, where
is a local ring which admits only a finite number of ideals for each
. Hence, if
, then we know from Lemma 2.3 that
. This is in contradiction to the assumption that
. Thus
is impossible. If
and
for some
, then we know from Lemma 2.2(v) that
. Moreover, if
and
is a field for each
, then it follows from
of Visweswaran and Patel (Citation2015, Theorem 3.2) that
. Thus
is also impossible. Therefore, either R has a unique maximal N-prime of (0) or has exactly two maximal N-primes of (0).
Proposition 4.3 characterizes rings R such that R has exactly two maximal N-primes of (0) satisfying the property that . We know from Visweswaran and Patel (Citation2015, Theorem 3.1) that such rings are necessarily Artinian. We use Example 4.2 in the proof of Proposition 4.3.
Example 4.2
Let and
be a SPIR such that
but
for each i. Let
. Then
.
Proof
It is clear that the vertex set of equals
. Observe that the subgraph of
induced on
is a clique. Moreover in
, the set of all neighbors of
equals
, the set of all neighbors of
equals
, and the set of all neighbors of
equals
. It follows from the above description of
that
.
Proposition 4.3
Let R be an Artinian ring which admits exactly two maximal ideals. Then the following statements are equivalent:
(i) |
| ||||
(ii) | Either |
Proof
We know from (Atiyah & Macdonald, Citation1969, Theorem 8.7) as rings, where
is a local Artinian ring for each
. Since
, it is clear that at least one between
and
cannot be a field. Moreover, it follows from Lemma 2.2(iii) and (Atiyah & Macdonald, Citation1969, Proposition 2.8) that
is principal for each
. Suppose that
is not a field whereas
is a field. Let
,
, and
. Then
as rings, where
is a SPIR and
is a field. From
, it follows from Proposition 2.7 that
but
and moreover,
. Suppose that both
and
are not fields. Then
for each
. Moreover,
is a SPIR and as
, it follows from Lemma 2.2(ii) that
for each
. Moreover, in this case, it follows from Example 4.2 that
.
If
as rings, where
is a SPIR with
but
and
is a field, then it follows from Proposition 2.7 that
. If
as rings, where
is a SPIR with
but
for each
, then we obtain from Example 4.2 that
.
We next try to classify rings R such that R admits exactly one maximal N-prime of (0) and . Let
denote the unique maximal N-prime of (0) in R. In Proposition 4.5, we present a classification of such rings R under the assumption that
for each
. We use
of Lemma 4.4 in the proof of Proposition 4.5.
Lemma 4.4
Let be a local Artinian ring such that
is not principal. Let
be such that
with
,
but
. The following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Note that and
.
Observe that
is an independent set of
. If
, then there exist
such that
. As
,
,
for each
, and since
, it follows that
for each
. This implies that
for each
and so
. This is impossible as
and
. Therefore,
.
Suppose that
. Since
, it follows that
and so
. As
, we obtain that
and hence,
. Therefore,
. Let
. Note that
and
. It is not hard to verify that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, it follows that
. As
, it is clear that
is an isolated vertex of
. Observe that
is the union of the clique H, where H is the subgraph of
induced on W with
and the isolated vertex
. Hence,
.
Suppose that . Now
. Let
. Note that
. Hence,
and
. Observe that
and
. It is easy to verify that the set of all nonzero proper ideals of R equals
. It is clear that the subgraph
of
induced on
is a clique on four vertices and the subgraph
of
induced on
is a clique on four vertices, and
is the union of
,
, and the isolated vertex
. Note that
and hence,
and
are not adjacent in
. Let
be a set of four distinct colors. If we assign the color
to
, color
to
, color
to
, and the color
to the vertices
,
, and
, then it is clear that the above assignment of colors is indeed a proper coloring of the vertices of
and moreover, it is evident from the above discussion that
.
The moreover assertion is already verified in the proof of .
Proposition 4.5
Let R be a ring which admits as its unique maximal N-prime of (0). Suppose that
for each
. Then the following statements are equivalent:
(i) |
| ||||
(ii) | R is necessarily a local Artinian with |
Proof
By hypothesis,
. Hence, it follows from Visweswaran and Patel (Citation2015, Lemmas 2.1 and 2.2) that
and it is clear that
. Therefore, there exist
such that
. We are assuming that
for each
. Hence,
cannot be principal. Moreover, as
,
, it follows from Remark 2.5 that
and
. Therefore, it follows that
and
are not comparable under inclusion for each
. Note that
is an independent set of
. From
, it follows that
and hence,
. It follows from
that
. As
and
, we obtain that
. Let
. Observe that
for each
. Moreover, if
, then
and
. Hence,
is an independent set of
. This is impossible. Therefore,
. This proves that
. Therefore,
is necessarily a maximal ideal of R and as
, we obtain that R is a local Artinian ring with
as its unique maximal ideal.
If (ii) holds, then the local Artinian ring
satisfies the hypotheses of Lemma 4.4 and in addition
. Hence, it follows from
of Lemma 4.4 that
.
If either (i) or (ii) holds, then again it follows from the proof of of Lemma 4.4 that
and
.
We mention some examples in Example 4.6 to illustrate Lemma 4.4.
Example 4.6
With the help of Belshoff and Chapman (Citation2007), Corbas and Williams (Citation2000a,Citation2000b), we mention some examples of finite local rings R such that and
.
(i) | Let | ||||
(ii) | Let | ||||
(iii) | Let | ||||
(iv) | Let |
Let R be a ring which admits as its unique maximal N-prime of (0). Suppose that
for some
. If
, then we know from Visweswaran and Patel Citation2015, Remark 3.5(ii)) that R is a local Artinian ring with
as its unique maximal ideal and moreover, we know from Lemma 3.4 that
can be generated by at most two elements. In Proposition 4.7, we classify the local Artinian rings R such that
under the assumption that
is principal. In such a case,
is a SPIR.
Proposition 4.7
Let be a SPIR. Then the following statements are equivalent:
(i) |
| ||||
(ii) | Either |
Proof
This follows immediately from Visweswaran and Patel (Citation2015, Proposition 3.7).
Let be a local Artinian ring such that
is not principal but
for some
. Our goal was to classify such local Artinian rings R satisfying the condition that
. First we assume that
, but
and present the desired classification in Propositions 4.9 and 4.10. In Proposition 4.9, we assume that
and in Proposition 4.10, we assume that
. We use Lemma 4.8 in the proof of Proposition 4.10.
Lemma 4.8
Let be a local Artinian ring such that
is not principal but
. Suppose that
, whereas
. If
, then
.
Proof
It is clear from the hypotheses that . Suppose that
. Then there exist
such that
and
. Thus there exist
such that
for each
and
. As
, and
is not principal, it follows that
is an independent set of
. This implies that
which contradicts the assumption that
. Therefore,
.
Proposition 4.9
Let be a local Artinian such that
is not principal but
. Suppose that
. If
, then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Note that
. Observe that
is a basis of
as a vector space over
. Therefore, the ideals
are distinct. Since
, we obtain that
is an independent set of
. If
, then
. In such a case,
is an independent set of
. This implies that
and this contradicts (i). Therefore,
. We next prove that
. If
, then there exists
such that
. Then
is an independent set of
. This is impossible as
. Therefore,
. Note that
. Thus
, and
. Hence,
, and
. Therefore,
, and
.
Let
. Note that
. It can be easily shown that the set of all nonzero proper ideals of R equals
. It is clear that
is an independent set of
. Hence,
. We next verify that
. Let
be a set of four distinct colors. Since
and
, it follows that the set of all isolated vertices of
equals
. Observe that
are not adjacent in
and both are not adjacent to
in
. Hence, if we assign the color
to
, color
to
, color
to
, color
to
, and the color
to all the vertices in the set
the set of all isolated vertices of
, then it is clear that the above assignment of colors is a proper vertex coloring of
. Therefore, we obtain that
and so
.
The moreover assertion that is already verified in the proof of
and the assertion that
is verified in the proof of
.
Proposition 4.10
Let be a local Artinian ring such that
is not principal but
for some
. Suppose that
, whereas
. If
, then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Note that and
.
We know from Lemma 4.8 that
. If
, then
. In such a case, it follows from Lemma 3.11 that
. This contradicts (i). Therefore,
.
Observe that and
. Hence,
and so
. Therefore,
.
Note that
. Observe that
vary over
. It is not hard to verify that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, it follows that
. Since
and
, it follows that
is the set of all isolated vertices of
. Note that
is the union of H and A, where H is the subgraph of
induced on W. It is clear from the above discussion that
.
The moreover assertion that is already verified in the proof of
and the assertion that
is verified in the proof of
.
In Example 4.11, we provide some examples to illustrate Propositions 4.9 and 4.10.
Example 4.11
With the help of theorems proved by Corbas and Williams (Citation2000a,Citation2000b) and from the examples of rings of order 32 given in Belshoff and Chapman (Citation2007], we give some examples of finite local rings of order 32 such that
satisfies the hypotheses of Proposition 4.9.
(i) | Let | ||||
(ii) | Let | ||||
(iii) | Let | ||||
(iv) | Let | ||||
(v) | Let |
Next with the help of Proposition 4.10, we give some examples of finite local rings with
such that
.
(a) | Let | ||||
(b) | Let | ||||
(c) | Let |
Let be a local Artinian ring such that
is not principal but
for some
. We next focus on classifying such rings
in order that
under the additional assumption that
, whereas
. The results regarding their classification are presented in Propositions 4.13, 4.16, and 4.18. We use Lemma 4.12 in the proof of Proposition 4.13.
Lemma 4.12
Let be a local Artinian ring such that
is not principal but
. Suppose that
, whereas
. If
, then either
or
Proof
Suppose that and
. If
, then
for some
. Note that r must be in
and so
for some
. Hence,
. This is in contradiction to the assumption that
. Therefore,
. Similarly, from the assumption that
, it follows that
. It follows from
is a basis of
as a vector space over
and
that the ideals
are distinct. Moreover, it follows from
, and
that
is an independent set of
. This implies that
and this contradicts the hypothesis that
. Therefore, either
or
.
Proposition 4.13
Let be a local Artinian ring such that
is not principal but
. Suppose that
,
, and moreover,
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Since
, it follows from Lemma 4.12 that
and
are comparable under the inclusion relation. By hypothesis,
. Hence, we can assume without loss of generality that
but
. Thus
for some
. Let
for some
. Then
. From
, it follows that
and so
. From
and
, it follows that
. Hence,
. It is now clear that
if and only if
. Suppose that
. Observe that
is an independent set of
. This implies that
and this is a contradiction. Therefore,
. If
, then there exists
such that
. Note that
is an independent set of
. This contradicts (i). Therefore,
.
Note that and as
, and by hypothesis,
. Hence,
, and
. Thus
, and so
.
We can assume without loss of generality that
but
. It follows as in
that
for some
. From
, it is clear that
and hence, u is a unit in R. As
,
for some
and so
. Note that
vary over
.
It is easy to verify that the set of all nonzero proper ideals of R equals . Since
is an independent set of
, it is clear that
. We next verify that
. Let
be a set of four distinct colors. Since
, it is clear that
is an isolated vertex of
. From
and
, it follows that in
,
and
are not adjacent,
and
are not adjacent, and
and
are not adjacent. If we assign the color
to
and
, color
to
and
, color
to
and
, and the color
to
and
, then it is not hard to verify that the above assignment of colors is indeed a proper vertex coloring of
. The above discussion shows that
. Therefore,
.
The moreover assertion that is already verified in the proof of
and the assertion that
is verified in the proof of
.
We illustrate Proposition 4.13 with the help of some examples in Example 4.14.
Example 4.14
With the help of results from Belshoff and Chapman (Citation2007) , Corbas and Williams (Citation2000a,Citation2000b), we mention some examples of local rings of order 32 satisfying the hypotheses of Proposition 4.13.
(i) | Let | ||||
(ii) | Let | ||||
(iii) | Let |
Let be a local Artinian ring satisfying all the hypotheses of Proposition 4.13 except the hypothesis
. That is, with the assumption that
, in Propositions 4.16 and 4.18, we investigate on the desired classification. In Lemma 4.15, we provide a necessary condition on
in order that
.
Lemma 4.15
Let be a local Artinian ring such that
is not principal but
. Suppose that
, and
. If
, then
.
Proof
Since R is local and , it follows from
that
for some unit
. From
, it is now clear that
. We consider the following cases:
Case(i).
In this case, we show that . Suppose that
. Then
. Hence, we can find
such that
for each
and moreover,
. Observe that
is an independent set of
and hence,
. This contradicts the hypothesis that
. Therefore,
.
Case (ii). but
Note that for some
and so
. From
, we obtain that
. In this case, we show that
. From
, it follows that
. If
, then there exists
such that
. Hence,
. Observe that
is an independent set of
. This implies that
and this is a contradiction. Therefore,
and so
.
Case(iii). and
In this case, we show that . Suppose that
. Note that
. Now there exists
such that
. Note that
is an independent set of
. This is impossible since by hypothesis,
. Therefore,
.
Proposition 4.16
Let be a local Artinian ring such that
is not principal but
. Suppose that
, and
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
It is clear that and
.
We know from the proof of Lemma 4.15 (see case(i) and case(ii) considered in the proof) that
. Since
, it follows from Lemmas 3.11 and 3.13 that
.
Note that and
. Hence,
. Therefore,
and so
.
Let
be such that
with
. Let
. Observe that
vary over A
. It is not hard to verify that the set of all nonzero proper ideals of R equals
. Note that
is an independent set of
. Hence,
. We next verify that
. As
, it is clear that
is an isolated vertex of
. From
and
, it follows that in
,
and
are not adjacent,
and
are not adjacent. Let
be a set of four distinct colors. Let us assign the color
to
and
, color
to
, color
to
and
, and the color
to
and
. Then it is clear that the above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
.
The moreover assertion that is already verified in the proof of
and the assertion that
is verified in the proof of
.
We illustrate Proposition 4.16 with the help of an example in Example 4.17.
Example 4.17
Let . Let
, where I is the ideal of T given by
. Observe that the unique maximal ideal
satisfies the hypotheses of Proposition 4.16 with
,
and moreover,
. Therefore, we obtain from the proof of
of Proposition 4.16 that
.
Proposition 4.18
Let be a local Artinian ring such that
is not principal but
for some
. Suppose that
, and
for some unit
with
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Note that and
. We are assuming that
for some unit u in R such that
. Hence, it follows that
.
It follows from Lemma 4.15 that
. Therefore,
.
Assume that . We prove that
. Suppose that
. From
, it follows that
. Let
. Note that
with
, and
. In such a case, it follows from
of Proposition 4.16 that
. This is a contradiction. Hence,
.
Note that and
. Hence,
if
,
if
, and
if
.
We first assume that
. Note that
. Let
. Observe that
for some
and so
. It is clear that
vary over A
. It is easy to verify that the set of all nonzero proper ideals of R equals
. Observe that
is an independent set of
. Hence,
. We next verify that
. Since
,
is an isolated vertex of
. From
, it follows that
and
are not adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
and
, color
to
, color
to
, and the color
to
and
. It is clear that the above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
.
Suppose that . In this case there exists
such that
and
. Note that either
for some
or
for some
. Hence, either
or
. Observe that
and
. Therefore, either
or
. In the case
, we obtain with
and
that
,
, and
. Therefore, we obtain from the proof of
of Proposition 4.16 that
. If
, then the elements
are such that
,
, and
. Hence, again it follows from the proof of
of Proposition 4.16 that
.
We next assume that . Then
. By hypothesis,
for some unit u in R such that
. Hence,
. Let
. Note that
vary oner A
. It is easy to show that the set of all nonzero proper ideals of R equals
. As
, it follows that
is an independent set of
. Hence,
. We next verify that
. Since
, it follows that
and
are not adjacent in
. As
, we obtain that
is an isolated vertex of
. It follows from
that in
,
and
are not adjacent and
and
are not adjacent. Let
be a set of four distinct colors. Let us assign the color
to
and
, color
to
and
, color
to
and
, and the color
to
and
. Note that the above assignment of colors is indeed a proper vertex coloring of
. This shows that
. Therefore,
.
The moreover assertion is already verified in the proof of
and the assertion that
is verified in the proof of
.
In Example 4.19, we illustrate Proposition 4.18 with the help of some examples.
Example 4.19
(i) | Let | ||||
(ii) | Let |
Lemma 4.20
Let be a local Artinian ring such that
is not principal but
for some
. Suppose that
, and
. If
, then the following hold:
(i) |
| ||||
(ii) |
|
Proof
(i) Since is a basis of
as a vector space over
, it follows that the ideals
are distinct. Moreover, we obtain from the hypotheses on the elements
that
is an independent set of
. We assert that
. Suppose that
. Then
. For if
, then
. Hence,
for some
. This implies that
and so
. Therefore,
. In such a case, it is clear that
, and moreover,
is an independent set of
. This is in contradiction to the hypothesis that
. Therefore,
.
Suppose that . Then there exists
such that
and
. Since
is an independent set of
and as
, it follows that either
or
. Hence,
and from
, we obtain that
. Therefore,
.
(ii) We consider two cases.
Case( I ).
If , then there exist
such that
. Note that
and as
, it follows that either
or
. Similarly, it follows that either
or
. Suppose that
. Since
and as
, it follows that
cannot be 0 and
cannot be 0. Similarly, if
, then
cannot be 0 and
cannot be 0. This is a contradiction. Therefore,
.
Case( II ).
Note that and as
, it follows that
cannot be an independent set of
. As
, it follows that
. Thus
. We assert that
. Suppose that
. Then there exist
such that
. Then either
or
. Without loss of generality, we can assume that
. Hence,
and
. Observe that
is an independent set of
. This is impossible. Therefore,
.
Proposition 4.21
Let be a loccal Artinian ring satisfying the hypotheses mentioned in Lemma 4.20. If
and
, then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
Note that from the given hypotheses on the elements
of
, we obtain that
. Since
,
is a vector space over
. We claim that
is linearly independent over
. Let
be such that
. We first assert that at least one of
. Suppose that
and
. Since
, it follows that
for some
. Now from
and
, it follows that
. This is a contradiction. Thus either
or
. If
, then from
, it follows that
. If
, then from
, we obtain that
. This shows that
is linearly independent over
. Therefore,
. Note that
, and so
. Hence, we obtain that
.
It is clear that
and
. Let
. Observe that
vary over A
. It is not hard to verify that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, it follows that
. We next verify that the vertices of
can be properly colored using a set of four distinct colors. As
, it follows that
is the set of all isolated vertices of
. It follows from
and
that no two members from
are adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
, color
to all the members from B, color
to
, and the color
to all the members from
. Observe that the above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
.
The moreover assertion that is verified in the proof of
.
With the help of results from Belshoff and Chapman (Citation2007), Corbas and Williams (Citation2000a,Citation2000b), we now mention some examples to illustrate Proposition 4.21.
Example 4.22
(i) Let and
, where I is the ideal of T given by
. Note that
is a local Artinian ring with
. Observe that
satisfies the hypotheses of Lemma 4.20 with
and
. Moreover,
and
. As
, it follows from the proof of
of Proposition 4.21 that
.
(ii) Let and I be the ideal of T given by
. Let
. Note that
is a local Artinian ring with
and moreover,
satisfies the hypotheses of Lemma 4.21 with
and
. Furthermore,
and
. As
, we obtain from the proof of
of Proposition 4.21 that
.
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.20. Suppose in addition that
and
. In Propositions 4.24 and 4.26, we provide the classification of such rings R in order that
. We use Lemma 4.23 in the proof of Propositions 4.24 and 4.26.
Lemma 4.23
Let be a local Artinian ring satisfying the hypotheses mentioned in Lemma 4.20. Suppose that
,
, and
. Then
. Moreover, if
, then
.
Proof
Note that is an independent set of
. We know from Lemma 4.20(i) that
. As
, from
, it follows that
and to prove
, it is enough to show that
. Suppose that
. Observe that
is an independent set of
. This is impossible since by hypothesis,
. Therefore,
.
(ii) We now prove the moreover assertion of the lemma. Note that . We assert that
is linearly independent over
. Let
be such that
. We assert that both a and b belong to
. First we verify that either
or
. If
and
, then
for some
. Therefore,
. This implies that
. Hence,
and from
, it follows that
. This is impossible since
. Thus either
or
. If
, then we get that
. As
, it follows that
. If
, then we obtain that
. If
, then
for some
. As
, it follows that
. This contradicts the hypothesis that
. Hence,
. This proves that
is linearly independent over
. Therefore,
.
Proposition 4.24
Let be a local Artinian ring satisfying the hypotheses mentioned in Lemma 4.20. Suppose that
, and
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
We know from Lemma 4.20(i) that
. We know from Lemma 4.23 that
. Note that
. Observe that
. Thus
, and
. Hence,
, and
. Thus
and so
.
From
,
, and
, it is clear that
. Observe that
vary over
. It is not hard to verify that the set of all nonzero proper ideals of R equals
. As
is an independent set of
, it follows that
. We next verify that
.. Since
is an isolated vertex of
. As
and
, no two members from
are adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
, color
to all the vertices from A, color
to
, and the color
to
and
. It is now evident that the above assignment of colors is indeed a proper vertex coloring of
. Hence, it follows that
. Therefore,
.
The moreover assertion that is verified in the proof of
.
Wet mention an example in Example 4.25 from Belshoff and Chapman (Citation2007) to illustrate Proposition 4.24.
Example 4.25
Let and I be the ideal of T given by
. Let
. Let
. Note that
is a local Artinian ring and
satisfies the hypotheses of Lemma 4.20 with
and
. Moreover,
and
. As
and
, it follows from the proof of
of Proposition 4.24 that
.
Proposition 4.26
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.20. Suppose that
, and
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
We know from Lemma 4.20(i) that
. Moreover, we know from Lemma 4.23 that
and
. Observe that
. Thus
and
. Hence,
and so
.
Note that
and so
. From
, and
, it follows that
and
. As
, we obtain that
is linearly independent over
. Let
. Now it is clear that
vary over A
. We now determine the set of all nonzero proper ideals of R. Let I be any nonzero proper ideal of R. Note that either
or
. We first consider the case in which
. Since
, it follows that
. We next consider the case in which
. Let
be such that
. Note that
for some
with at least one between a and b equals 1. If
, then it is easy to see that
. Hence, either
or 2. If
, then
. If
, then
. In such a case, it is clear that
. Suppose that
and
. Note that
. Let us denote the ideal Rr by C. Observe that R / C is a local Artinian principal ideal ring with
as its unique maximal ideal. As
, it follows from
of (Atiyah & Macdonald, Citation1969, Proposition 8.8) that
is the set of all proper ideals of R / C. Therefore,
. It is easy to verify that
. It follows from the above arguments that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, it follows that
. We next verify that
. Since
,
, and
, it is clear that the subgraph of
induced on
is a clique. That is, no two members from
are adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
, color
to all the vertices from
, color
to
, and the color
to
. Note that the above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
.
Example 4.27 illustrates Proposition 4.26.
Example 4.27
Let and I be the ideal of T given by
. Let
. Note that
is a local Artinian ring with
satisfies the hypotheses of Lemma 4.20 with
and
. Moreover,
and
. As
, and
, it follows from the proof of
of Proposition 4.26 that
.
Proposition 4.28
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.20. Suppose that
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
We know from the moreover part of Lemma 4.20(i) that
. Also with
, it follows from the proof of the moreover part of Lemma 4.20(i) that
. Thus
is an one-dimensional vector space over
. Hence,
. Note that
and so
. Therefore,
.
From
,
, it is clear that
. Hence,
. Since
and
, we obtain that
. With
, we have
vary over A
. It is easy to verify that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, it follows that
. We next verify that
. As
,
is an isolated vertex of
. Observe that
and
are not adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
and
, color
to
, color
to
, and the color
to
and
. It is clear that the above assignment of colors is indeed a proper vertex coloring of
. This shows that
. Therefore,
.
The moreover assertion that is verified in the proof of
.
We illustrate Proposition 4.28 with the help of Example 4.29.
Example 4.29
Let and I be the ideal of T given by
. Let
. Note that with
,
is a local Artinian ring and it satisfies the hypotheses of Lemma 4.20 with
and
. Moreover,
, and
. Therefore, it follows from the proof of
of Proposition 4.28 that
.
Proposition 4.30
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.20. Suppose that
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
We know from the moreover part of Lemma 4.20(i) that
. From
, it follows that
, for some
such that
and
. It follows from the proof of the moreover part of Lemma 4.20(i) that either
or
. Hence,
and so
. From
, we obtain that
. Therefore,
.
As in the proof of
, we can assume that
. Now it is clear that
and
. Therefore,
. From
, it follows that either
or
. Without loss of generality, we can assume that
. Let
. Note that
vary over A
. It is easy to verify that the set of all nonzero proper ideals of R equals
. Since
is an independent set of
, we obtain that
. We next verify that
. As
, it is clear that
is an isolated vertex of
. It follows from
that in
,
and
are not adjacent and
and
are not adjacent. Let
be a set of four distinct colors. Let us assign the color
to
and
, color
to
, color
to
and
, and the color
to
and
. It is clear that the above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
..
The moreover assertion that is verified in the proof of
.
In Example 4.31, we provide an example to illustrate Proposition 4.30.
Example 4.31
Let and let I be the ideal of T given by
, where
. Let
. Note that
is such that
is a local Artinian ring and it satisfies the hypotheses of Lemma 4.20 with
, and
. Moreover,
, and
. Therefore, we obtain from the proof of
of Proposition 4.30 that
.
Lemma 4.32
Let be a local Artinian ring such that
is not principal, but
,
, and
. If
, then
. Moreover,
and
are principal.
Proof
First we show that . By contrary, suppose that
. Then either
or
. Without loss of generality, we can assume that
. Note that
is an independent set of
. It is clear that
. From
,
, and
, it follows that
. This implies that
and so
. This is a contradiction. Therefore,
.
We next verify that is principal. It follows from the hypotheses on
that
is an independent set of
. We claim that
. If
, then
and moreover,
is an independent set of
. This is in contradiction to the assumption that
. Hence,
and so
for some
. From
, it follows that
. Therefore,
. Similarly, we obtain that
and so
. Note that
. We first verify that
is principal. It is clear that
. If either
or
, then it follows that
is principal. Suppose that
and
. Then
is an independent set of
. It follows from
that
. Hence,
. This proves that
is principal. Similarly, it can be shown that
is principal. From
, it follows that
is principal if either
or
. If
and
, then
is an independent set of
. It follows from
that
. Hence , we obtain that
and therefore,
is principal.
We next show that is principal.. Now
for some
. Hence,
. If either
or
, then it is clear that
is principal. Suppose that
and
. In such a case,
is an independent set of
. Since
, we obtain that
. Hence,
is principal.
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.32 and if in addition, suppose that
is not principal. In Lemma 4.33, we provide some necessary conditions in order that
.
Lemma 4.33
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.32. In addition, suppose that
is not principal. If
, then the following hold:
(i) |
| ||||
(ii) |
|
Proof
It is clear from the hypotheses that is an independent set of
. It is noted in the proof of Lemma 4.32 (see the second paragraph of its proof) that
and
.
(i) Let be such that
. Then
. Since
, it follows that
must be adjacent to at least one member of W in
. As
is not principal by assumption, it follows that
. This implies that
. If
is such that
, then a similar argument yields that
. Hence, we arrive at
. Since
is not principal, it follows that
. This proves that
.
(ii) We first show that . Suppose that
. As
, it follows that
. Note that
is an independent set of
. As
and since
, it follows from the hypotheses on
that
. From the assumption that
, it follows that
and moreover,
. This implies that
is an independent set of
. This is in contradiction to the assumption that
. Therefore,
. Similarly, it follows that
. We next verify that
. Suppose that
. Then U is an independent set of
. From
, it follows that
. This implies as before that
. Thus
. Similarly, it can be shown that
. If
, then we obtain that
is an independent set of
. It follows from
that
. This implies that
since
. This proves that
.
With the hypotheses on as in Lemma 4.33, in Proposition 4.34, we provide a classification of rings R such that
under the additional assumptions that
is not principal and
.
Proposition 4.34
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.33. In addition, suppose that
is not principal and
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
It is already noted in the first paragraph of the proof of Lemma 4.33 that
is an independent set of
and moreover, it is shown there that
. And it follows from Lemma 4.33(i) that
.
Suppose that and
. In such a case, as
, it follows that
. Thus if
, then we obtain that
is an independent set of
. This is in contradiction to the assumption that
. Hence,
. Similarly, it follows that
.
Suppose that . As
is not principal, it follows that
and
. If
, then
is an independent set of
. This contradicts the assumption that
. Therefore,
. Note that
. Thus if
, then
is an independent set of
. This is a contradiction. Therefore, we obtain that
. It follows from a similar argument that
. Using the fact that
is an independent set of
, it follows as argued above that
.
Suppose that
. Since
,
is a vector space over
. From
and
is not principal, it follows that
. Note that
. Hence,
and so
. Let
. Observe that
vary over A
. If
, then with
, we get that
and moreover,
, whereas
and
. Hence, we obtain from the proof of
of Proposition 4.21 that
. So we can assume that
. Then
,
, and
. As the subgraph H of
induced on
is a clique, it follows that
. We next verify that
. Note that as
, any nonzero ideal I of R with
is an isolated vertex of
. Let
be the set of all isolated veertices of
. It can be easily verified that the set of all nonzero proper ideals B of R with
equals
. Hence,
is the union of H and
. It is now clear that
.
Suppose that . As
is a two-dimensional vector space over
, it follows that
. Observe that
and so
. Therefore,
. Let
. Note that
vary over A
. Since
is an independent set of
, it follows that
. We next verify that
. As
, each nonzero ideal I of R with
is an isolated vertex of
. Let
be the set of all isolated vertices of
. It is easy to verify that the set of all proper nonzero ideals of R which are not contained in
equals
. From
, it is clear that
and
are not adjacent in
. Let
be a set of four distinct colors. Let us assign the color
to
, color
to
, color
to
and
, and the color
to
and all the vertices from
. The above assignment of colors is indeed a proper vertex coloring of
. This proves that
. Therefore,
.
The moreover assertion regarding |R| is verified in the proof of and the assertion that
is verified in the proof of
.
We illustrate Proposition 4.34 with the help of some examples in Example 4.35.
Example 4.35
(i) Let and
, where I is the ideal of T given by
. Note that R is a local Artinian ring with
as its unique maximal ideal. Observe that with
, it is clear that
. Moreover,
,
,
.,
is not principal,
, and
. Now it follows from the proof of
of Proposition 4.34 that
. This example is found in Belshoff and Chapman (Citation2007, p. 479).
(ii) Let and
, where I is the ideal of T given by
. Note that R is a local Artinian ring with
as its unique maximal ideal. Let
and
. It is clear that
,
,
,
. Moreover, note that
is not principal,
,
, but
, and
. It follows from the proof of
of Theorem 4.34 that
. This example is found in Belshoff and Chapman(Citation2007, p. 479).
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.32.. Suppose that
but
and
is principal. In Propositon 4.37, we classify such rings R in order that
. Lemma 4.36 determines
in order that
.
Lemma 4.36
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.32. Suppose that
is principal and
. If
, then
.
Proof
Assume that . We know from Lemma 4.32 that
. Moreover, it is shown in the proof of Lemma 4.32 (see the second paragraph of iits proof) that
and
and using them, it is verified there that
. Observe that
. Similarly, it follows from
that
. Hence,
and
. It follows from
that
and therefore, we obtain that
. From
, it is clear that
and
. It follows from
that
. By assumption,
is principal. Let
be such that
. It now follows that there exist units
such that
and
. Therefore, we obtain that there exist units
such that
, and
. Thus
.
We next verify that . It follows from
that
is an independent set of
. Observe that for any
,
. Since
, it follows that
cannot form an independent set of
. Therefore,
. Let
. It follows from
that
. From
, we obtain that
. This proves that
.
Proposition 4.37
Let be a local Artinian ring satisfying the hypotheses of Lemma 4.32. Suppose that
is principal,
but
. Then the following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof
We know from Lemma 4.36 that
. Moreover, we know from the proof of Lemma 4.36 that there exist units
such that
and
. It follows from
that if w is any unit in R, then
. Hence,
for some
and so
. Similarly, it follows that
. Moreover, it is noted in the proof of Lemma 4.36 that there exist units
such that
. Thus
. It follows from
and
that if
, then
. Therefore,
and so
. Moreover,
. Let
and
. Observe that
. Note that
,
,
,
,
, and
. We now verify that
. Observe that
, and
. Therefore,
and
. Thus
and so
.
If (ii) holds, then it is clear that
with
satisfies the hypotheses of Proposition 4.24 and (ii) of Proposition 4.24. Therefore, it follows from
of Proposition 4.24 that
.
The moreover assertion that follows from the proof of
of Proposition 4.24.
In Example 4.38, we mention an example to illustrate Proposition 4.37.
Example 4.38
Let and
, where I is the ideal of T given by
. Observe that R is a local Artinian ring with
as its unique maximal ideal. Note that
satisfies the hypotheses of Lemma 4.32 with
and
. Moreover, it is clear that
is principal,
, and
. Furthermore,
satisfies (ii) of Proposition 4.37 with
and
. Therefore, we obtain from
of Proposition 4.37 that
, and it follows from the moreover part of Proposition 4.37 that
. This example is found in Belshoff and Chapman (Citation2007) and is already mentioned in this article (see Example 4.25).
Acknowledgements
We are very much thankful to the referee for a careful reading of our article and for many valuable and useful suggestions. We are very much thankful to Professor Jonas Hartwig for the support.
Additional information
Funding
Notes on contributors
S. Visweswaran
S. Visweswaran is serving as a professor in the Department of Mathematics, Saurashtra University, Rajkot, India. His research area is Commutative Ring Theory.
Jaydeep Parejiya
Jaydeep Parejiya is a lecturer in the Department of Mathematics, Government Polytechnic, Rajkot. He has registered for his PhD in the Department of Mathematics, Saurashtra University. His research interest is in Commutative Ring Theory.
References
- Aalipour, G., Akbari, S., Behboodi, M., Nikandish, R., Nikmehr, M. J., & Shaveisi, F. (2014). The classification of the annihilating-ideal graphs of commutative rings. Algebra Colloquium, 21, 249. doi:10.1142/S1005386714000200
- Aalipour, G., Akbari, S., Nikandish, R., Nikmehr, M. J., & Shaiveisi, F. (2012). On the coloring of the annihilating-ideal graph of a commutative ring. Discrete Mathematics, 312, 2620–2625.
- Anderson, D. F., Axtell, M. C., & Stickles, Jr., J. A., (2011). Zero-divisor graphs in commutative rings. In M. Fontana, S. E. Kabbaj, B. Olberding, & I. Swanson (Eds.), Commutative algebra, Noetherian and non-Noetherian perspectives. New York, NY: Springer-Verlag.
- Anderson, D. F., Frazier, A., Lauve, A., & Livingston, P. S. (2001). The zero-divisor graph of a commutative ring II. Lecture Notes in Pure and Applied Mathematics, 220, 61–72.
- Anderson, D. F., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217, 434–447.
- Anderson, D. D., & Naseer, M. (1993). Beck’s coloring of a commutative ring. Journal of Algebra, 159, 500–514.
- Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Reading, MA: Addison-Wesley.
- Axtell, M., Coykendall, J., & Stickles, J. (2005). Zero-divisor graphs of polynomial and power series over commutative rings. Communications in Algebra, 33, 2043–2050.
- Balakrishnan, R., & Ranganathan, K. (2000). A textbook of graph theory (Universitext). New York, NY: Springer.
- Beck, I. (1988). Coloring of commutative rings. Journal of Algebra, 116, 208–226.
- Behboodi, M., & Rakeei, Z. (2011a). The annihilating-ideal graph of commutative rings I. Journal of Algebra and Applications, 10, 727–739.
- Behboodi, M., & Rakeei, Z. (2011b). The annihilating-ideal graph of commutative rings II. Journal of Algebra and Applications, 10, 741–753.
- Belshoff, R., & Chapman, J. (2007). Planar zero-divisor graphs. Journal of Algebra, 316, 471–480.
- Corbas, B., & Williams, G. D. (2000a). Rings of order p\textsuperscript{5} I, Nonlocal rings. Journal of Algebra, 231, 677-690.
- Corbas, B., & Williams, G. D. (2000b). Rings of order p\textsuperscript{5} II, Local rings. Journal of Algebra, 231, 691–704.
- Hadian, M. (2012). Unit action and geometric zero-divisor ideal graph. Communications in Algebra, 40, 2920–2930.
- Heinzer, W., & Ohm, J. (1972). On the Noetherian-like rings of E.G. Evans. Proceedings of the American Mathematical Society, 34, 73–74.
- Kaplansky, I. (1974). Commutative rings. Chicago, IL: University of Chicago Press.
- Lucas, T. G. (2006). The diameter of a zero-divisor graph. Journal of Algebra, 301, 173–193.
- Smith, N. O. (2003). Planar zero-divisor graphs. Journal of Commutative Rings, 2, 177–186.
- Visweswaran, S. (2011). Some properties of the complement of the zero-divisor graph of a commutative ring. ISRN Algebra, 2011, 24 p., Article ID 59104.
- Visweswaran, S., & Patel, H. D. (2015). On the clique number of the complement of the annihilating ideal graph of a commutative ring. Contributions to Algebra and Geometry.