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Abstract
Let α be an endomorphism of an arbitrary ring with identity. The aim of this article is to introduce the ideal
of
as an α-skew M-McCoy which is a generalization of α-rigid ideals and McCoy ideals and to investigate its properties. α-skew M-McCoy is an α-skew McCoy ideals relative to a monoid
. The findings show that if
for some positive integer
and
be an α-skew M-McCoy left ideal of
then
is α-skew M-McCoy left ideal of
. Also if
be a locally finite ring and
be an α-skew M-McCoy left ideal of
, then
is α-semicommutative left ideal of
.
PUBLIC INTEREST STATEMENT
Let α be an endomorphism of an arbitrary ring R with identity. In this article we introduce ideal I of R is α-Skew M-McCoy (α-Skew McCoy ideals related to a monoid) which are a generalization of McCoy ideals and investigate their properties. We do this by considering the α-Skew McCoy condition on polynomials in R[M, α] instead of R[M]. This provides us with an opportunity to study skew McCoy ideals in a general setting and several known results on skew McCoy ideals that are obtained.
1. Introduction
Throughout this paper, denotes an associative ring with unity. In (Nielsen, Citation2006), Nielsen introduced the notion of a McCoy ring. A ring
is said to be right McCoy (resp., left McCoy) if for each pair of nonzero polynomials
with
, then there exists a nonzero element
with
(resp.,
). A ring
is McCoy if it is that left and right McCoy. Kamal paykan in (Paykan & Mussavi, Citation2017) express definition of right
- McCoy
-
satisfy
, then there exists
such that
. where
is a monoid, and
a monoid homomorphism. The ring
is called weak
- skew McCoy with respect to
if for any nonzero polynomials
and
in
with
, there exists
such that
for
(Nikmehr, Nejati, & Deldar, Citation2014). Aghapouramin and Nikmehr (Citation2018) introduced the notion of a nil-M-McCoy ring. A ring
is right nil-M-McCoy, if whenever
,
, with
,
satisfy
, then
for some nonzero
and for each
. Left nil-M-McCoy rings are defined similarly. If
is both left and right nil-M-McCoy, then
is nil-M-McCoy. According to Nikmehr (Citation2014), a left ideal
of
is called
-skew Armendariz if the following conditions are satisfied:
1. For any and
implies
,
2. if and only if
.
Aghayev, Harmanci, and Halicioglu (Citation2010) defined a ring that is called
-abelian if, for any
, and any idempotent
,
and
if and only if
and a ring
is called
-semicommutative if, for any
,
implies
and
if and only if
. According to Krempa (Citation1996), an endomorphism
of a ring
is called rigid if
implies
for
. We call a ring
-rigid if there exists a rigid endomorphism
of
(see Esmaeili & Hashemi, Citation2012). There are many ways to generalize the McCoy condition, and we direct reader to the excellent papers (Chen & Tong, Citation2005; Cui & Chen, Citation2012; Habibi & Mousavi, Citation2012; Mousavi, Keshavarz, Rasuli, & Alhevaz, Citation2011; Song, Liand, & Yang, Citation2011; Zhao & Liu, Citation2009) for nice introduction to these topics. Given the fact that
-skew McCoy rings has been studied, but its connection with the monoid has not been investigated yet. Therefore the present study tries to fill the gap.
We are motivated to introduce the notion of -skew M-McCoy left ideal
of
with respect to an endomorphism
of
. This notion extends both ideal skew McCoy ideals an
-skew McCoy ideals. We do this by considering the
-skew McCoy condition on polynomials in
instead of
. This provides us with an opportunity to study skew McCoy ideals in a general setting and several known results on skew McCoy ideals that are obtained. Constructing various examples, we classify how the
-skew McCoy ideals relative to a monoid
property behaves under various ring extensions.
2. α-Skew McCoy ideals related to a monoid
We begin this section by the following two definitions and also we study properties of -skew M-McCoy left ideals.
Definition 2.1 Let be an endomorphism, a left ideal
of
is called
-skew McCoy if the following are satisfied:
1. For any and
implies
i) ideal of
is a left
-skew McCoy if there exists a nonzero element
with
for all
,
ii) ideal of
is a right
-skew McCoy if there exists a nonzero element
with
for all
.
2. if and only if
.
If ideal of
is both left
-skew McCoy and right
-skew McCoy, then we say that ideal
of
is an
-skew McCoy.
Definition 2.2 Let be a monoid and let
be an endomorphism of a ring
, a left ideal
of
is called
-skew M-McCoy if the following satisfied:
1. For any and
such that
and
i) ideal of
is a left
-skew M-McCoy if there exists a nonzero element
with
, for all
,
ii) ideal of
is a right
-skew M-McCoy if there exists a nonzero element
with
for all
.
2. if and only if
.
If ideal of
is both left
-skew M-McCoy and right
-skew M-McCoy, then we say that ideal
of
is an
-skew M-McCoy.
Example 2.1 Let be a monoid and let
be an
-skew M-McCoy and consider
where
and
are the sets of all integers and all rational numbers, it is clear that
is the left ideal of
. Let
be an automorphism defined by
. Then
(1) is not
-rigid since
, but
if
.
(2) I of is
-skew M-McCoy left ideal.
let and
, where
,
for
,
such that
. Let
,
,
,
. Since
thus for any
, that
,
, thus
. Since
is
-skew M-McCoy, left ideal, hence
for all
,
and
. If set
, then
. Since
is arbitrary, thus
. Therefore,
. If
. We have
for all
and
for all
. Now we consider
,
,
,
. Thus
and hence . Since
is
-skew M-McCoy,
. Thus
Therefore, is an
-skew M-McCoy left ideal.
One may conjecture that a ring may be right ideal
-skew M-McCoy when
and
are both right ideal
-skew M-McCoy rings for any nonzero proper ideal
of
, where
is considered as a ring without identity. However, the answer is the following:
Example 2.2 Let be a field and consider
, then
is not right
-skew M-McCoy (hence not right ideal
-skew M-McCoy) by (Hirano, Hong, & Kim, Citation2014, Remark 2.2(2)). Note that all nonzero proper ideals of
are
We will show that and
are both right ideal
-skew M-McCoy for any nonzero ideal
of
. First, let
. Then
is right ideal
-skew M-McCoy. Let
such that
and
where
and
for
,
. We can write
and
where
,
and
,
. From
we have
and
if
, then
and so
a contradiction. Thus
and
, i.e.,
. This yields that
for every nonzero
, in the ideal of
generated by
. Next let
. Then
is right ideal
-skew M-McCoy. Let
for
and
, and
, where
and
for
,
, since
or
, we get
from
, entailing
. This yields that
for every nonzero
, in the ideal of
generated by
. Finally let
. Then
is right
-skew M-McCoy.
is clearly right ideal
-skew M-McCoy, since
. Similarly we show that
is left ideal
-skew M-McCoy. Now we consider
,
,
,
. Thus
, and hence
, since
is
-skew M-McCoy,
. Thus
if and only if
. Therefore,
is an
-skew M-McCoy left ideal.
The following example shows that there exists a ring such that
is
-skew M-McCoy. But
and
is not
-skew M-McCoy.
Example 2.3 Let be the ring of integers and
be the ring of integers modulo 4. Consider a ring
Let be an endomorphism defined by
Then is
-skew M-McCoy by the similar method in the proof of (Hong, Kim, & Kwak, Citation2003, Proposition 15). However, for an ideal
of
, the factor ring
is not -skew M-McCoy. In fact,
but
Recall that if is an endomorphism of a ring
, then the map
defined by
is an endomorphism of the polynomial ring
and clearly this map extends
. We shall also denote the extended map
by
and the image of
by
.
Theorem 2.1 Let be a monoid and
be an endomorphism such that
for some positive integer
. And
be a left ideal of
. If
is an
-skew M-McCoy, then
is
-skew M-McCoy.
Proof. Let ,
with
where
we need to prove each
, and
for some nonzero
.□
Assume that for each
and
, where
and
. Take a positive integer
such that
for any
and
, where the degree is as polynomial in
and the degree of zero polynomial is taken to be 0.
Suppose that ,
. Then
. And the set of coefficient of
(resp.,
) equal the set of coefficients of
(resp.,
). It is easy to check that
, since
and
. Since
is
-skew M-McCoy, there exists
such that
and
. And
implies
for any
, and
. Hence
for any
.
Therefore is left
-skew M-McCoy,
implies
for any
and
, since
. Hence we have
for any . Therefore
is right
-skew M-McCoy. Now we consider
,
,
. Thus
. Since
is
-skew M-McCoy,
. Thus
if and only if
. Therefore
is
-skew M-McCoy. The proof is completed.□
Recall that an element in
is called regular if
, i.e.,
is not a zero divisor. For subrings of an
-skew M-McCoy ring, we have the following.
Proposition 2.1 Let be a monoid and
be an endomorphism of a ring
and
be an ideal of
satisfying that every nonzero element in
is regular. If
is
-skew M-McCoy, then
is
-skew M-McCoy (without identity).
Proof. Let ,
with
, since
is an ideal of
and
is
-skew M-McCoy, there exists nonzero elements
satisfying
for any
, and
for any
. Therefore
for any nonzero element
(otherwise, if
(resp.,
) for a element
, then
(resp.,
. Hence
(resp.,
) since every nonzero element in
is regular. This is a contradiction). Consequently, we have
for any , and
. Thus
is left (resp., right)
-skew M-McCoy. Now we consider
,
,
. Thus
. Since
is
-skew M-McCoy,
. Thus
. Therefore
is an
-skew M-McCoy left ideal.□
Theorem 2.2 Let be a monoid and
be a ring and
be left ideals of
. If
and
an
-skew M-McCoy left ideal of
,then
is an
-skew M-McCoy left ideal of
.
Proof. Let and
such that
. Then
. Thus there exists
such that
and
, hence
and
.
Now we consider ,
,
. Thus
. Since
is
-skew M-McCoy,
. Thus
if and only if
. Therefore,
is an
-skew M-McCoy.□
The following is an immediate corollary of Theorem 2.2:
Corollary 2.1 Let be a ring and
an left ideal of
. If
is
-skew M-McCoy then
is an
-skew M-McCoy ring.
To prove Proposition 2.4, we state the following propositions:
Proposition 2.2 If is an
-skew M-McCoy left ideal of
and for some
and some integer
,
and
then
.
Proof. Consider ,
,
. Since
is an
-skew M-McCoy left ideal of
, so
.□
Recall that a left ideal of
is called
-abelian if, for any
and any idempotent
, we have the following conditions:
1. ,
2. if and only if
.
Next, we show that every -skew M-McCoy left ideal of
is an
-abelian left ideal.
Proposition 2.3 If is an
-skew M-McCoy left ideal of
, then
is an
-abelian left ideal.
Proof. Assume that is an
-skew M-McCoy left ideal of
. Consider
and let
,
,
with
. Then clearly
and
and hence
and then by Proposition 2.3,
. So
. Let
,
, and
, we also have
. So
then
.□
Recall that a left ideal of
is called
-semicommutative if, for any
we have the following conditions:
1. then
,
2. if and only if
.
A ring is called locally finite if every finite subset of
generated a finite semigroup multiplicatively. Finite rings are clearly locally finite and the algebraic of a finite field is locally finite but it is not finite.
Proposition 2.4 Let be a locally finite ring and
be an
-skew M-McCoy left ideal of
. Then
is
-semicommutative left ideal of
.
Proof. Let with
. For any
, since
is locally finite there exists integers
such that
. So we obtain inductively
, put
then
with
. Notice that
. Hence,
is an idempotent and so by Proposition 2.3,
and
. Thus,
. On the other hand by Proposition 2.3,
, so
, and hence
. Since
is an
-skew M-McCoy left ideal of
so
, and by Proposition 2.2, we imply that
for all
.□
Let be an endomorphism of a ring
and
be the
matrix over ring
and
defined by
. Then
is an endomorphism of
. It is obvious that, the restriction of
to
is an endomorphism of
, where
is the
diagonal matrix ring over
. We also denote
by
.
Proposition 2.5 Let be an endomorphism of a ring
. Then
is an
-skew M-McCoy left ideal of
if
is an
-skew M-McCoy left ideal for any
.
Proof. Let and
satisfying
, where
and
.
Then from , it follows that,
for each . Since
is an
-skew M-McCoy left ideal of
, then there exists
such that
and
for any
and
. Therefore
Now we consider ,
,
Thus , and hence
. Since
is an
-skew M-McCoy,
. Thus
if and only if
.
Therefore is an
-skew M-McCoy left ideal.□
Additional information
Funding
Notes on contributors
Vahid Aghapouramin
Vahid Aghapouramin is a assistant professor, at the department of mathematics, Bonab Branch, Islamic Azad university, Bonab, Iran. His research interests are pure mathematics, Commutative Algebra, Noncommutative Algebra.
Mohammad Javad Nikmehr
Mohammad Javad Nikmehr is a associate professor, Faculty of Mathematics, K.N. Toosi, University of Technology, P.O. Box 16315- 1618, Tehran, Iran. His research interest are Commutative Algebra, Graph Theory, Noncommutative Algebra, Algebraic Combinatorics.
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