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Abstract
This article concerns skew polynomial rings over Armendariz rings and -skew Armendariz ring. Let R be a Noetherian, Armendariz, prime ring. In this paper we prove that R and the polynomial ring R[x] are 2-primal. Further we prove that if
is an endomorphism of a ring R, then (1) R is a
-skew Armendariz ring implies that
is a
-skew Armendariz ring, where
is an extension of
to
. (2) R is a
-rigid implies that
is a 2-primal.
AMS Subject Classifications:
Public Interest Statement
One of the earliest examples in non-commutative algebra was skew polynomial rings also known as Ore extensions. Skew polynomial rings have invited attention from Mathematicians and in this area considerable work has been done and investigations are on.
The characterization of ideals and prime ideals (in particular associated prime ideals, completely prime ideals and minimal prime ideals), and 2-primal property of Ore extensions has lead to the extension of certain notions from commutative setup to non-commutative setup. Ore extensions constitute an important class of rings, appearing in extensions of differential calculus, in non-commutative geometry, in quantum groups and algebras and as a uniting framework for many algebras appearing in physics and engineering models.
1. Introduction and preliminaries
A ring R always means an associative ring with identity , unless otherwise stated. Let
be an endomorphism of ring R. The skew polynomial ring or Ore extension of endomorphism type is denoted by
. The prime radical and the set of nilpotent elements of R are denoted by P(R) and N(R) respectively. The ring of integers is denoted by
, unless otherwise stated.
We begin with the following:
A ring R is called 2-primal if the prime radical of R coincides with the set of nilpotent elements of R i.e. or if the prime radical is completely semi-prime (see Birkenmeier, Heatherly, & Lee, Citation1993 for more details). An ideal I of a ring R is called completely semi-prime if
implies
for
. We also note that any reduced ring is 2-primal and a commutative ring is also 2-primal. The class of 2-primal rings is closed under subrings by Birkenmeier et al. (Citation1993, Proposition 2.2).
Example 1.1
Let . Here
and any nilpotent element of R is of the form
. Hence R is 2-primal.
Example 1.2
Let . R is not a reduced ring as
is a non-zero nilpotent element and hence R is not 2-primal.
Krempa (Citation1996) has investigated the relation between minimal prime ideals and completely prime ideals of a ring R. With this he proved the following:
Theorem 1.3
For a ring R the following conditions are equivalent:
(1) | R is reduced. | ||||
(2) | R is semiprime and all minimal prime ideals of R are completely prime. | ||||
(3) | R is a subdirect product of domains. |
According to Krempa (1996) an endomorphism of a ring R is called rigid if implies that
for
. We call a ring
-rigid if there exists a rigid endomorphism
of R.
Example 1.4
Let . Then
defined as
is an endomorphism of R. Further implies that
i.e.
which gives
. Hence
. Thus R is a
-rigid ring.
Example 1.5
(Bhat, Citation2011, Example 1) Let , where F is a field. Let
an automorphism be defined by
Let . Then
.
But .
Hence R is not a -rigid ring.
Also R is said to be -compatible if for each
,
implies and is implied by
. Also a ring R is
-rigid if and only if R is
-compatible and reduced. Moreover, R is
-rigid if and only if
is reduced (Hong, Kim, & Kwak, Citation2003, Proposition 3).
Example 1.6
Let D be an integral domain. Consider the commutative ring
Let be an automorphism of R defined by
where u is a fixed element of D. Then R is -compatible.
Example 1.7
Let be a commutative ring, where
is the ring of integers modulo 2. Let
be defined by
Then is an automorphism of R. Now
. But
. Hence R is not
-compatible.
2. Armendariz rings
The notion of Armendariz rings was introduced by Rege and Chhawchharia (Citation1997). They defined a ring R to be an Armendariz ring if whenever polynomial
satisfy , then
for each i, j. (The converse is always true.) This ring was named so because as Armendariz (Citation1974, Lemma 2) had noted that a reduced ring satisfies this condition. In addition to reduced rings, quotient rings over a commutative P.I.D. are Armendariz (Rege & Chhawchharia, Citation1997, Theorem 2.2). But every
full matrix ring over any ring is not Armendariz, where
(Rege & Chhawchharia, Citation1997). Note that Armendariz rings are defined through polynomial rings over them. Also subrings of Armendariz rings are Armendariz. Anderson and Camillo (Citation1998) has found a relation between an Armendariz ring and reduced ring as:
Theorem 2.1
(Anderson & Camillo, Citation1998, Theorem 7) If R is a prime ring which is left and right Noetherian, then R is Armendariz if and only if R is reduced.
With this we prove the following:
Theorem 2.2
Let R be a Noetherian Armendariz prime ring. Then R is 2-primal.
Proof
By Theorem (2.1), R is a reduced ring. We know that a reduced ring is 2-primal. Hence R is 2-primal.
The converse is not true.
Example 2.3
Let . Then R is a commutative ring and hence 2-primal. Let
. Now
Hence R is not an Armendariz ring.
Theorem 2.4
Let R be a Noetherian prime ring. If R is an Armendariz ring, then P(R) is completely semi-prime.
Proof
As proved in Theorem (2.1), R is a reduced ring and by using Theorem (1.3), the result follows.
The converse of the above is not true.
Example 2.5
Let F be a field, . Here P(R) is completely semi-prime, as R is a reduced ring. Let
,
. Then
But . Hence R is not an Armendariz ring.
Concerning polynomial rings over some kinds of rings, we have the following results:
(1) | A ring R is reduced if and only if R[x] is reduced. | ||||
(2) | A ring R is 2-primal if and only if R[x] is 2-primal (Birkenmeier et al., Citation1993, Proposition 2.6). | ||||
(3) | A ring R is abelian if and only if R[x] is abelian (Kim & Lee, Citation2000, Theorem 8). Note that a ring R is said to be abelian if every idempotent of it is central. |
Theorem 2.6
(Anderson & Camillo, Citation1998, Theorem 2) A ring R is Armendariz if and only if R[x] is Armendariz.
Hilbert’s Basis Theorem (1890) states that:
Theorem 2.7
(Goodearl & Warfield, Citation2004, Theorem 1.9) If R is a Noetherian ring, then R[x] is a Noetherian ring.
These help us to prove a relation between Armendariz rings and 2-primal rings as:
Theorem 2.8
Let R be a Noetherian Armendariz prime ring. Then R[x] is 2-primal.
Proof
By Theorem (2.6), R[x] is Armendariz. Also using Hilbert’s Basis Theorem, it follows that R[x] is a Noetherian ring. Therefore, by Theorem (2.2), R[x] is 2-primal.
The converse is not true.
Example 2.9
Let F be a field. Let . Then
and any nilpotent element of R is of the form
. Hence R is 2-primal. By Proposition (2.6) of Birkenmeier et al. (Citation1993), R[x] is 2-primal. Let
and
Now , but
. Hence R is not an Armendariz ring.
3. ![](//:0)
-skew Armendariz rings
Recall that is the usual polynomial ring with coefficients in R, in which multiplication is subject to the relation
for all
. We take any
to be of the form
. By Anderson and Camillo (Citation1998, Theorem 2), polynomial rings over Armendariz rings are also Armendariz. There is a natural motivation to investigate the nature of skew polynomial ring over a Armendariz ring, but the fact is that a skew polynomial ring over an Armendariz ring need not be Armendariz as follows:
Example 3.1
(Kim & Lee, Citation2000, Example 6) Let be the ring of integers modulo 2 and consider the ring
with the usual addition and multiplication. Then R is a commutative reduced ring; hence R is Armendariz by (Citation1974, Lemma 1). Now let
be defined by
Then is an automorphism of R. We claim that
is not Armendariz. Let
Then , but
. Therefore,
is not an Armendariz ring.
We now discuss -skew Armendariz rings (
an endomorphism of a ring R) and their extensions. Recall (Hong et al, Citation2003) that a ring R with an endomorphism
is called a
-skew Armendariz ring if for
implies that
, for all
and
. It is also known as skew Armendariz ring with endomorphism
. Every subring of a
-skew Armendariz ring is
-skew Armendariz.
Example 3.2
(Hong et al., Citation2003, Example 1) Consider the commutative ring
Let be an automorphism of R defined by
Then R is a -skew Armendraiz ring.
Example 3.3
Let F be a field and a ring. Define an endomorphism
by
For ,
, we have
. But
. Hence R is not a
-skew Armendariz ring.
Let be an endomorphism of ring R. Then
can be extended to an endomorphism (say
) of
by
We now prove the following Theorem:
Theorem 3.4
Let R be a ring, an endomorphism of R such that R is a
-skew Armendariz ring. Then
is a
-skew Armendariz ring.
Proof
Let
be such that . Then
(3.1)
(3.1)
Let be such that
. Let
and
where
. Let
Then
and
and the set of coefficients of the (res.,
equals the set of coefficients of the
(res.,
. Now
and
,
. Since R is
-skew Armendariz using Equation (3.1), the result follows.
Also from Hong et al. (Citation2003):
Theorem 3.5
(Hong et al., Citation2003, Proposition 3) Let be an endomorphism of R. Then
is reduced if and only if R is
-rigid.
With this we prove the following:
Theorem 3.6
Let be an endomorphism of ring R such that R is a
-rigid. Then
is a 2-primal.
Proof
By Theorem (3.5), is reduced and hence it is 2-primal.
The converse is not true.
Example 3.7
Consider a commutative polynomial ring over . Let
and
be an endomorphism defined by
.
Then
(1) | R is 2-primal as | ||||
(2) | R is | ||||
(3) | R is not |
We now give an example of a 2-primal ring which is not a -skew Armendariz ring.
Example 3.8
Let where
and
are reduced rings. Then
is 2-primal, because
. Let
be an endomorphism defined by
Take and
. Then
. But
. Hence
is not a
-skew Armendariz ring. Note that
here is not rigid.
Additional information
Funding
Notes on contributors
V.K. Bhat
Meeru Abrol received the MPhil degree in Mathematics from University of Jammu, India. Currently, she is an associate professor at Govt. Women College, Ghandhi Nagar, Jammu, India. Her research area is Algebra (ring theory). Her work is supported under QIP of UGC, Govt. of India.
Meeru Abrol
V.K. Bhatreceived the PhD degree in Mathematics from University of Jammu, India. Currently, he is a professor at Shri Mata Vaishno Devi University, Katra, India. Bhat is a researcher in the area of Algebra (ring theory) and theoretical computer science. He has published about 80 research papers in refereed journals. He has guided eight research students for their PhD degree. He has completed two research projects funded by Department of Atomic Energy, Govt. of India and University Grants Commission, Govt. of India. He has attended a number of conferences and research schools in various countries.
References
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