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Abstract
In this article, we prove some results for lower nil -Armendariz ring. Let
be a strictly totally ordered monoid and
be a semicommutative ideal of
. If
is a lower nil
-Armendariz ring, then
is lower nil
-Armendariz. Similarly, for above
, if
is 2-primal with
and
is
-Armendariz, then
is a lower nil
-Armendariz ring. Further, we find that if
is a monoid and
a u.p.-monoid where
is a
-primal
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Public Interest Statement
In this manuscript, we proved some results for lower nil - Armendariz rings which is introduced by Alhevaz and Hashemi in 2016. In fact, they discussed in their work about Upper nilradicals and Lower Nilradicals of unique product monoid rings in connection with the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. In general,
, but for lower nil
-Armendariz ring with semicommutative ring
, we proved that
is 2-primal ring. As the consequences, when ring
is a lower nil
-Armendariz, then ring of triangular matrices and skew upper triangular matrices are lower nil
-Armendariz rings. The logical relationship among above-mentioned notions and other significant classes of Armendariz like rings can be helpful to provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings.
1. Introduction
Throughout this article, denotes an associative ring with identity unless otherwise stated. For a ring
,
,
and
denote the prime radical (lower nilradical), upper nilradical and the set of nilpotent elements of
, respectively. It is known that
. Here,
denotes the polynomial ring with an indeterminate
over
and
for the set of all coefficients of
.
and
represent the full matrix ring and upper triangular matrix ring of order
over the ring
, respectively.
A ring is said to be Armendariz ring if two polynomials
, such that
implies
for each
,
. A ring
is reduced if it has no nonzero nilpotent element. Armendariz (Citation1974) himself proved that every reduced ring is satisfying above condition. Later, the term Armendariz ring was coined by Rege and Chhawchharia (Citation1997). A ring
is said to be semicommutative by Narbornne (Citation1982), whenever
implies
for
and it is 2-primal if
. Birkenmeier et al. (Citation1993, Proposition 2.2) showed that the class of 2-primal rings is closed under subrings. A ring
is called
, by Marks (Citation2001), if
.
Let be a monoid and
the identity element of
. Then,
denotes the monoid ring of
over the ring
. Due to Liu (Citation2005), a ring
is called
-Armendariz ring, whenever elements
,
satisfy
, then
for each
and
. Clearly, every ring is an
-Armendariz ring, where
. But, if
is a semigroup with trivial multiplication
for all
and
, the semigroup
with identity, then none of the ring is an
-Armendariz. If
, then
is
-Armendariz if and only if
is Armendariz ring. Also, a monoid
with
and
is
-Armendariz, then
is a p.p.-ring if and only if
is a p.p.-ring.
Alhevaz and Moussavi (Citation2014) said a ring to be a nil
-Armendariz if
,
such that
implies
for each
and
. Later, Alhevaz and Hashemi (Citation2017) introduced the concept of upper and lower nil
-Armendariz ring relative to a monoid. A ring
is upper
nil
-Armendariz ring if whenever elements
,
such that
, then
for each
and
.
2. Lower nil m-armendariz rings
Recall that a monoid is said to be u.p. monoid (unique product monoid) if for any two nonempty finite subsets
of
, there exists an element
uniquely presented in the form of
where
and
. The class of unique product monoids is quite large and important. For example, this class contains the right or left ordered monoids, torsion-free nilpotent groups, submonoids of a free group. Usefulness of unique product monoids and groups are extensively studied relating to the zero divisor problems. The ring theoretical property of unique product monoid has been established by many researchers in past, we refer Cheon and Kim (Citation2012); Liu (Citation2005); Okninski (Citation1991); Passman (Citation1977).
Proposition 2.1. For a unique product monoid , every
-primal ring is a lower nil
-Armendariz ring.
Proof. Let be a unique product monoid. Take
,
such that
. Then
in
. Since
is reduced, therefore by Liu (Citation2005, Proposition 1.1),
is
-Armendariz ring. This implies
for each
and
. Thus,
is a lower nil
-Armendariz ring.□
Corollary 2.1. For any unique product monoid , semicommutative ring is a lower nil
-Armendariz ring.
A monoid equipped with an order “
” is said to be an ordered monoid if for any
,
implies
and
. Moreover, if
implies
and
, then
is said to be strictly totally ordered monoid.
Since each strictly totally ordered monoid is a u.p. monoid, hence by Proposition 2.1, we have the following result.
Corollary 2.2 Let be a strictly totally ordered monoid. Then every
-primal ring is a lower nil
-Armendariz ring.
The following example shows that the condition u.p. monoid in Proposition 2.1 is not superfluous.
Example 2.1. Let , where
is the unit matrix of
for each
. Then
is a monoid but it is not a u.p. monoid. Let
and
. Then
but
. Hence,
is a not a lower nil
-Armendariz ring.
Proposition 2.2. Let be a u.p. monoid,
a lower nil
-Armendariz ring and
, a subring of
.
(1) If , then
is a lower nil
-Armendariz ring.
(2) If is an NI ring, then
is a lower nil
-Armendariz ring.
Proof. (1) Let , where
. Then by assumption,
. Since
is a lower nil
-Armendariz ring, therefore
for each
. Also,
, so
and hence
for each
.
(2) Since is NI,
. Also, for lower nil
-Armendariz ring, we have
. Therefore,
. Thus,
is a lower nil
-Armendariz ring.□
Proposition 2.3. For a u.p. monoid , every lower nil
-Armendariz ring is nil
-Armendariz ring.
Proof. We have, is a lower nil
-Armendariz ring if and only if
is a
-Armendariz ring. Then, by using Alhevaz & Hashemi, Citation2017, Lemma 3.1(c)) and Hashemi (Citation2013, Theorem 2.23), we obtain the result.□
But converse is not true. In this regard, we have an example.
Example 2.2. Suppose is a u.p. monoid and consider the ring given in Hwang, Jeon, and Lee (Citation2006, Example 1.2). Let
be a reduced ring,
a positive integer and
. Each
is an NI ring by Hwang et al. (Citation2006, Proposition 4.1(1)). Define a map
by
, then
can be considered as subring of
via
. Notice that
, with
whenever
, is a direct system over
. Set
be the direct limit of
. Then
, and
is an NI by Hwang et al. (Citation2006, Proposition 1.1). By Alhevaz and Moussavi (Citation2014, Theorem 2.2), NI ring is nil
-Armendariz ring. By Jeon, Kim, Lee, and Yoon (Citation2009, Theorem 2.2(1)),
is a semiprime ring, hence
. Here
. Thus, by Alhevaz & Hashemi, Citation2017, Lemma 3.1(c)),
is not a lower nil
-Armendariz ring.
Recall that a monoid is cancellative if for all
,
implies
and
.
Proposition 2.4. Let be an ideal of a cancellative monoid
. If
is a lower nil
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. Suppose ,
such that
. Take
, then
and
,
when
. Now,
Since is a lower nil
-Armendariz ring, therefore
for each
. Thus,
is a lower nil
-Armendariz ring.□
Definition 2.1. Let be a ring and
for a multiplicative closed subset
consisting of all central regular elements of
. Then
is a ring.
Proposition 2.5. Let be a monoid and
a multiplicatively closed subset of the ring
consisting of central regular elements. Then
is a lower nil
-Armendariz ring if and only if so is
.
Proof. Let be a lower nil
-Armendariz ring and
where
and
. Here, we consider
and
, where
for each
and
. From
, we have
. Since
is the lower nil
-Armendariz ring, therefore
. Moreover,
, so
. Thus,
is a lower nil
-Armendariz ring.□
Converse is obvious, since and
is a subring of
. Therefore,
is a lower nil
-Armendariz ring.
Lemma 2.1. Let be a monoid with at least one non trivial element of finite order and
be a ring such that
. Then
is not a lower nil
-Armendariz ring.
Proof. Suppose has order
and take
,
, then
i.e.
but
. Hence,
is not a lower nil
-Armendariz ring.□
Lemma 2.2. Let be a submonoid of the monoid
. If
is a lower nil
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Definition 2.2. If contains elements of finite order in an abelian group
, then
is a fully invariant subgroup of
. The group
is said to be a torsion-free group if
.
Theorem 2.1. Let be a finitely generated abelian group. Then the following conditions are equivalent:
(1) is torsion-free.
(2) There exists a ring with
such that
is a lower nil
-Armendariz ring.
Proof. If
is a finitely generated abelian group with
. Then
. By Liu (Citation2005, Lemma 1.13),
is a u.p.-monoid. Also, by Corollary 2.1,
is a lower nil
-Armendariz, for any semicommutative ring
with
.□
Let
and
. Then
is a cyclic group of finite order. If a ring
is a lower nil
-Armendariz ring, then
is lower nil
-Armendariz ring by Lemma 2.2, which contradicts Lemma 2.1. Thus, every ring
is not a lower nil
-Armendariz ring. Hence,
and
.
Proposition 2.6. For a monoid , if
is a semicommutative lower nil
-Armendariz ring, then
.
Proof. Let , where
for each
. Now,
for each
. Then there exists
, a positive integer, such that
for each fixed
, where
. Let
. Let
and
. Then
. For brevity of notation, let
. Then
. Here, in
, each term of
, there exist some
for
, at least
times. Therefore, we can replace
by
, where
and
is a product of some elements from the set
. Since
and
is a semicommutative ring, so
. Therefore,
and hence
. Thus,
.□
Conversely, let . Then, there exists a positive integer
such that
, for each
This implies,
i.e.,
. Since
is a lower nil
-Armendariz ring, therefore,
implies
. Hence,
for each
and this implies
. Thus,
.
Proposition 2.7. Let be a monoid and
a u.p. monoid. If
is a semicommutative lower nil
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. Here, every lower nil -Armendariz ring is a nil
-Armendariz ring. Also, by Alhevaz and Moussavi (Citation2014, Proposition 2.12),
and from Proposition 2.6,
. Again, by semicommutative of
, we get
. Hence,
is a
-primal ring. Finally, by Proposition 2.1, we conclude that
is a lower nil
-Armendariz ring.□
Theorem 2.2. Let be a monoid and
a u.p. monoid. If
is a semicommutative and lower nil
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. First note that defined by
is a ring isomorphism. Now, suppose
, where
,
and
for each
. In order to prove
for each
, let
and
. From Proposition 2.6, we have
. Further,
is a semicommutative ring and
a u.p. monoid, so by Corollary 2.1,
is a lower nil
-Armendariz ring. Again, by Cheon and Kim (Citation2012, Theorem 3), we have
. Hence,
. Therefore, by Proposition 2.6,
, for each
and
. Since
is a lower nil
-Armendariz ring, therefore
for each
, this implies
. Thus,
is a lower nil
-Armendariz ring.
A ring is said to be a Dedekind finite
- finite) if
implies
for each
.
Proposition 2.8. Let be a cyclic group of order
. Then each lower nil
-Armendariz ring is a Dedekind finite.
Proof. Let be a lower nil
-Armendariz ring and assume on the contrary
is not a Dedekind finite. Then by Goodearl (Citation1979, Proposition 5.5),
contains an infinite set of matrix units, say
. Consider the elements
and
of
. Then
but
is not a strongly nilpotent, which contradicts the assumption. Hence,
is a Dedekind finite.□
Theorem 2.3. Let be a monoid and
a u.p. monoid. If
is a semicommutative lower nil
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. By Liu (Citation2005, Theorem 2.3), we have and by Proposition 2.6,
. Rest part of the proof follows Proposition 2.7.
Let ,
, be monoids for index set
. Denote
there exist only finite
such that
the identity of
. Then
is a monoid under the binary operation
.□
Corollary 2.3. Let be the u.p. monoids and
a semicommutative ring. If
is a lower nil
-Armendariz ring for some
, then
is a lower nil
-Armendariz ring.
Proof. Let ,
such that
. Then
for some finite subset
. From Theorem 2.3 and by applying induction, the ring
is a lower nil
-Armendariz ring, therefore
. Hence,
is a lower nil
-Armendariz ring.□
Theorem 2.4. Let be a strictly totally ordered monoid and
an ideal of the ring
. If
is semicommutative and
is a lower nil
-Armendariz ring, then
is a lower nil
-Armendariz.
Proof. Let and
with
, where
. Then
. Now, we use transfinite induction on the strictly totally ordered set
to prove
, for each
. Clearly,
. Since
is a lower nil
-Armendariz ring, so there exists a positive integer
such that
for each
and
. Also,
if
or
. This implies that
. Now, suppose
, for any
and
with
. In order to prove
, for each
,
and
, consider
. Then
is a finite set. We write
as
such that
. Since
is a cancellative monoid,
and
imply that
. Also, since
is a strictly totally ordered monoid,
and
imply
. So, we have
. Now,
Note that, for any
and by induction hypothesis, we have
. Consider
for some positive integer
, then
. Therefore,
This implies
Continuing the above procedure, we get
This implies that . Similarly, we can see that
for
. Since
is a semicommutative ideal, so
is an ideal of
, therefore
. On the other hand, multiplying by
from right in Equation (2.2), we have
, so
and hence,
. Continuing this process, we get
for
. So,
for each
with
. Thus,
for each
.□
Proposition 2.9. Let be a monoid with
. Then the following conditions are equivalent:
(1) is a lower nil
-Armendariz ring.
(2) is a lower nil
-Armendariz ring.
Proof. First, we claim
For this, we consider ,
then for each
. So,
for each
,
. Then there exists a positive integer
corresponding to nilpotency of all elements of
such that
. Hence,
.□
Conversely, . Then
. This implies
and hence
for each
. Therefore
Next, we prove is a lower nil
-Armendariz ring. Since
defined by
is an isomorphism. Let ,
such that
, where
. Also, let
,
, for
Then by above isomorphism, it is easy to see that
. So
and this implies
for each
. Therefore,
. Thus,
is a lower nil
-Armendariz ring.
Proposition 2.10. Let be a monoid with
. Then the following condition are equivalent:
(1) is a lower nil
-Armendariz ring.
(2) is a lower nil
-Armendariz ring.
Proof. It is easy to see that there exists an isomorphism between rings
and
, defined by
Let and
such that
, where
We have . Therefore, from
, we get
, for
. Since
is a lower nil
-Armendariz ring, therefore
, for each
and all
,
and hence
for each
. Thus,
is a lower nil M-Armendariz ring.
Note that
is isomorphic to
subring of
. Clearly,
is a lower nil
-Armendariz ring.
Now, following example makes it clear that full matrix ring over a ring
is not a lower nil
-Armendariz ring.
Example 2.3. Let be a monoid with
and
is a ring. Take
and let
and
. Then
but
. Therefore,
is not a lower nil
-Armendariz ring.
Let be a ring with an endomorphism
such that
. Chen, Yang, and Zhou (Citation2006), considered the skew upper triangular matrix ring as a set of all upper triangular matrices with operations usual addition of matrices and multiplication subjected to the condition
, i.e. for any two matrices
and
, we have
, where
for each
and it is denoted by
.
It is noted that
The subring of the skew triangular matrices with constant main diagonal is denoted by . Also, the subring of skew triangular matrices with constant diagonals is denoted by
. We can denote
by
. Then
is a ring with addition is pointwise and multiplication defined by
where for each
. On the other hand, there exists a ring isomorphism
, defined by
, with
. So,
, where
is the skew polynomial ring with multiplication subject to the condition
for each
.
Also, we consider the following subrings of
Let be a monoid and
be an identity of
. Suppose
is a free monoid generated by
and
is a factor of
. Setting certain monomial in
to
, it is enough to show that for some
,
for any
. Let
be a ring with an endomorphism
. Then we can form the skew monoid ring
, by taking its elements to be finite formal combinations
, with multiplication subject to the relation
, for each
. It is easily seen that
and also the ring
and
fit naturally into
with
and
, respectively.
Theorem 2.5. Let and
be two monoids and
an endomorphism of the ring
with
. Then the following hold:
(1) is lower nil
-Armendariz if and only if so is
;
(2) is lower nil
-Armendariz if and only if so is
;
(3) is lower nil
-Armendariz if and only if so is
;
(4) is lower nil
-Armendariz if and only if so is
;
(5) is lower nil
-Armendariz if and only if so is
;
(6) is lower nil
-Armendariz if and only if so is
.
Proof. (1) Suppose that is a lower nil
-Armendariz ring. Let
and
such that
, where
for
and
. Then
, where
and
. Since
is a lower nil
-Armendariz ring, therefore,
for each
and hence
for each
. Thus,
is a lower nil
-Armendariz ring.□
Rest results can be easily proved by following above arguments.
Proposition 2.11 (1) Direct product of lower nil -Armendariz rings is a lower nil
-Armendariz ring.
(2) Direct sum of lower nil -Armendariz rings is a lower nil
-Armendariz ring.
Proof. (1) It is well known that . Let
and
such that
. Then
and
, where
and
. So,
where
. Since
is the lower nil
-Armendariz ring, therefore
. Thus,
is a lower nil
-Armendariz ring.
(2) It is easily seen that . As above, direct sum of lower nil
-Armendariz ring is a lower nil
-Armendariz ring.□
Theorem 2.6. The classes of lower nil -Armendariz rings are closed under direct limit.
Proof. Let be a direct system of lower nil
-Armendariz rings
for
and ring homomorphisms
for each
satisfying
, where
is a directed partially ordered set. Let
be the direct limit of
with
and
. Now, we will prove
is lower nil
-Armendariz ring. If we take
, then
,
for some
and there is
such that
,
. Define
where ,
. Then
forms a ring with
and
. Also, let
for
and
. Then there are
such that
,
,
,
and hence
. Since
is a lower nil
-Armendariz,
and therefore,
. Thus,
is a lower nil
-Armendariz ring.
Weak Annihilator: Let be a ring. Then for a subset
of
, the weak annihilator of
in
is
. If
is singleton, say
, then we use
. Also, for the ring
, we define
For an element
denotes the set of all coefficients of
and for a subset
of
,
denotes the set
.□
Now, we present the results of Alhevaz and Hashemi (Citation2017) and Cheon and Kim (Citation2012), which are helpful forour next result.
Theorem 2.7. Alhevaz and Hashemi (Citation2017, Theorem 3.2) Let be a u.p. monoid with nontrivial center. If
is a lower nil
-Armendariz ring, then
.
Theorem 2.8. Cheon and Kim (Citation2012, Theorem 3) Let be a ring and
a u.p. monoid. Then
.
Theorem 2.9. Cheon and Kim (Citation2012, Theorem 5) Let be a ring and
a u.p. monoid. Then
if and only if
.
Remark 2.1. Now, one can easily conclude that if is
-primal ring and
a u.p. monoid, then
.
Based on the above discussion, we have the following results:
Theorem 2.10. Let be a u.p. monoid with nontrivial center and
a lower nil
-Armendariz. Then
defined by for every
is bijective.
Proof. By Alhevaz and Hashemi (Citation2017, Theorem 3.2), we have and also from Proposition 2.3,
is a nil
-Armendariz ring. Therefore, by Lunqun and Jinwang (Citation2013, Theorem 3.1), result is true.
A ring is said to be nilpotent p.p. ring if for any
, the
is a right ideal generated by a nilpotent element of
.□
Theorem 2.11. Let be a monoid and
, a semicommutative lower nil
-Armendariz ring. If
is nilpotent p.p. ring, then so is
.
Proof. Let and
. Then
. Also, by Alhevaz and Moussavi (Citation2014, Proposition 2.12),
, therefore
and hence
for each
. Since
, so there exist at least one
,
such that
. Therefore,
being nilpotent
ring, there exist some
such that
.□
Now, we claim . Since
for each
, so
for some
. Therefore,
, hence
.
Also, for any , since
and
semicommutative ring, we have
for each
. So
. Hence,
. Thus,
, where
.
3. Lower nil ![](//:0)
-Armendarz rings from ![](//:0)
-Armendariz rings
Theorem 3.1. Let be a strictly totally ordered monoid and
a proper ideal of
with
If
is
-Armendariz and
, the 2-primal ring, then
is a lower nil
-Armendariz ring.
Proof. Suppose is an
-Armendariz ring and
is a 2-primal ring. Since,
, so
. Also,
is an ideal of
, so
and hence
. Since
is a 2-primal,
, therefore, it is reduced.□
Let and
such that
with
, where
. Now, we use transfinite induction on strictly totally ordered set
to show
. Since
, therefore
. Also,
is
-Armendariz, therefore
for each
and
. If there exist
and
such that
, then
and
. If
, then
implies
, a contradiction. Hence,
, similarly
. Thus,
.
Now, let be such that for any
,
,
. To prove
for any
and
with
, Consider
. Then
is a finite set. So, we put
such that
. Here,
is cancellative,
and
imply
. Since
strictly totally ordered monoid,
and
, therefore,
. Thus, we have
. Now
For any ,
and by induction hypothesis
. Take
, then
, since
for each
and
. Also,
, so
. Moreover, it is known that
is reduced, therefore,
. Again,
Multiplying (3.3) by from right, we get
. This implies
. Since
is reduced, therefore,
.
Again, by Equation (3.3), we have
so, . Multiplying Equation (3.5) by
from right side, we get
. This implies
, because
is reduced. Continuing the procedure, we get
for
with
. Therefore, by transfinite induction,
, for each
and
. Thus,
is a lower nil
-Armendariz ring.
Recall that a monoid is torsion-free if
and
are such that
implies
.
Corollary 3.1 Let be a commutative, cancellative and torsion-free monoid with
. If either one of the following conditions holds, then
is a lower nil
-Armendariz ring.
(1) is a 2-primal ring.
(2) is an
-Armendariz ring for some ideal
of
and
is
-primal with
.
Proof. If is commutative, cancellative and torsion-free monoid, then by Ribenboim (Citation1992, Result 3.3), there exists a compatible strict total order monoid “≤” on
. Then, by Corollary 2.2 and Theorem 3.1, results hold.
A ring is a right (resp. left) uniserial ring if its lattice of right (resp. left) ideals is totally ordered by inclusion. Right uniserial rings are also called right chain rings or right valuation rings because they are obvious generalization of commutative valuation domains. Like commutative valuation domains, right uniserial rings have a rich theory and they offer remarkable examples, we refer Marks, Mazurek, and Ziembowski (Citation2010).□
Proposition 3.1. (Alhevaz & Hashemi, Citation2017, Proposition 3.1(a)) For any u.p. monoid , every
-Armendariz ring is a lower nil
-Armendariz ring.
Proposition 3.2. For any u.p. monoid , every right or left uniserial ring is a lower nil
-Armendariz ring.
Proof. Let be a uniserial ring and
a u.p. monoid. By Marks et al. (Citation2010, Corollary 6.2),
is
-Armendariz ring. Therefore, by Proposition 3.1,
is a lower nil
-Armendariz ring.□
Proposition 3.3. Let be a monoid and
a u.p.-monoid. If
is semicommutative ring as well as
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. Let and
such that
. This implies
for each
and
, since
is
-Armendariz ring. Also
is semicommutative, therefore
for each
. Again, we can easily see that
. So from Corollary 2.1,
is a lower nil
-Armendariz ring.□
Lemma 3.1. Let be a monoid. If
is a
-primal
-Armendariz ring, then
is
-primal ring and
is a lower nil
-Armendariz ring with
.
Proof. By Lunqun and Jinwang (Citation2013, Theorem 2.3), we have is a nil
-Armendariz ring. Since
is
-primal, therefore
is a lower nil
-Armendariz ring. Also, by Lunqun and Jinwang (Citation2013, Lemma 2.2, Theorem 2.3), we have
and
is
-primal, so
is
-primal. Hence,
.□
Proposition 3.4. Let be a monoid and
a u.p.-monoid. If
is a
-primal
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. Since is a u.p.-monoid, therefore
, by Cheon and Kim (Citation2012, Theorem 3). Now, from Lemma 3.1,
. Next, from Lemma 3.1,
is a lower nil
-Armendariz ring. Further, rings
and
are isomorphic under the map
Now, let
where and
. We claim that
for each
and
. Let
Then
Threfore,
By Lemma 3.1, is a lower nil
-Armendariz ring, so
for all . Also, from Lemma 3.1,
is a lower nil
-Armendariz ring, hence
for each
. Therefore,
for each . Thus,
is a lower nil
-Armendariz ring.□
Proposition 3.5. Let be a monoid and
a u.p.-monoid. If
is
-primal
-Armendariz ring, then
is a lower nil
-Armendariz ring.
Proof. Construction of this proof is based on the proof of Liu (Citation2005, Theorem 2.3).□
Let . Without loss of generality, we assume
with
, where
.
Then
Note that for any
, when
. It is easy to see that there exists an isomorphism between rings
and
defined by
Assume
Then from the above isomorphism, we have
By Lemma 3.1, is a lower nil
-Armendariz ring. Therefore,
Also, by Lemma 3.1, is a lower nil
-Armendariz ring. Therefore,
for each
and
. Moreover,
for each
and
, where
and
. Hence,
is a lower nil
-Armendariz ring.
Example 3.1. Let be a ring and
a monoid. Then
is a
-primal
-Armendariz ring. Here,
. Let
be a monoid generated by elements
by the following defining relations:
Then by Okninski (Citation1991, Example 13 of Chapter 10), due to Krempa, the monoid is a u.p.-monoid. Therefore, by Proposition 2.1,
is a lower nil
-Armendariz ring. Also, by Liu (Citation2005, Theorem 2.3), we have
, hence
is a lower nil
-Armendariz ring.
Acknowledgements
The authors are thankful to Indian Institute of Technology Patna for providing the research facilities and to anonymous referees and the editor of this journal for their valuable suggestions to improve the presentation of the manuscript.
Additional information
Funding
Notes on contributors
![](/cms/asset/8a46525e-11de-47fe-92b0-7f355c024116/oama_a_1545411_ilg0001.jpg)
Sushma Singh
Sushma Singh completed her MSc in Mathematics from the Department of Mathematics, Banaras Hindu University, Varanasi, India, in 2010. Currently, she is a Ph. D. Research Scholar at the Department of Mathematics, Indian Institute of Technology, Patna, and her research area is Rings and Modules.
Om Prakash
Om Prakash is an associate professor at the Department of Mathematics, Indian Institute of Technology, Patna. He completed his MSc from Patna University, Patna, in 1999 and PhD from Banasthali University, Rajasthan, in 2010. He has 18 years of teaching experience in reputed institutions. His main research interest includes Rings and Modules, Algebraic Number Theory, Algebraic Coding Theory and Algebraic Graph Theory. He has the credit of published more than 30 research papers in international journals.
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