On α-skew McCoy ideals related to a monoid

Abstract

Let α be an endomorphism of an arbitrary ring $R$ with identity. The aim of this article is to introduce the ideal $I$ of $R$ 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 $M$. The findings show that if ${\alpha }^t=I_R$ for some positive integer $t$ and $I$ be an α-skew M-McCoy left ideal of $R$ then $I[M,\alpha ]$ is α-skew M-McCoy left ideal of $R[M,\alpha ]$. Also if $R$ be a locally finite ring and $I$ be an α-skew M-McCoy left ideal of $R$, then $I$ is α-semicommutative left ideal of $R$.

Keywords:

• α-skew McCoy ideals
• α-skew M-McCoy ideals
• α-semicommutative

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, $R$ denotes an associative ring with unity. In (Nielsen, 2006), Nielsen introduced the notion of a McCoy ring. A ring $R$ is said to be right McCoy (resp., left McCoy) if for each pair of nonzero polynomials $f(x),g(x)\in R[x]$ with $f(x)g(x)=0$, then there exists a nonzero element $r\in R$ with $f(x)r=0$ (resp., $rg(x)=0$). A ring $R$ is McCoy if it is that left and right McCoy. Kamal paykan in (Paykan & Mussavi, 2017) express definition of right $(M,\omega )$- McCoy $\alpha \beta \in R\ast M$-$\left\{0\right\}$ satisfy $\alpha \beta =0$, then there exists $0\ne r\in R$ such that $\alpha r=0$. where $M$ is a monoid, and $\omega :M\to End(R)$ a monoid homomorphism. The ring $R$ is called weak $\alpha$- skew McCoy with respect to $\alpha$ if for any nonzero polynomials $p(x)={\sum }_{i=0}^n{a}_i{x}^i$ and $q(x)={\sum }_{j=0}^m{b}_j{x}^j$ in $R[x;\alpha ]$ with $p(x)q(x)=0$, there exists $r\in R-\left\{0\right\}$ such that $a_i{\alpha }^i(r)\in nil(R)$ for $0\le i\le n$ (Nikmehr, Nejati, & Deldar, 2014). Aghapouramin and Nikmehr (2018) introduced the notion of a nil-M-McCoy ring. A ring $R$ is right nil-M-McCoy, if whenever $\alpha =a_1{g}_1+\cdots +a_n{g}_n$, $\beta =b_1{h}_1+\cdots +b_m{h}_m\in R[M]-\left\{0\right\}$, with $g_i,h_j\in M$, $a_i,b_j\in R$ satisfy $\alpha \beta \in nil(R)[M]$, then $a_{i}r\in nil(R)$ for some nonzero $r\in R$ and for each $1\le i\le n$. Left nil-M-McCoy rings are defined similarly. If $R$ is both left and right nil-M-McCoy, then $R$ is nil-M-McCoy. According to Nikmehr (2014), a left ideal $I$ of $R$ is called $\alpha$-skew Armendariz if the following conditions are satisfied:

1. For any $f(x)=\sum _{i=0}^n{a}_i{x}^i$ and $g(x)=\sum _{j=0}^m{b}_j{x}^j\in R[x,\alpha ],f(x)g(x)\in r_{R[x,\alpha ]}(I[x])$ implies $a_i{\alpha }^i(b_j)\in r_R(I)$,

2. $ab\in r_R(I)$ if and only if $a\alpha (b)\in r_R(I)$.

Aghayev, Harmanci, and Halicioglu (2010) defined a ring $R$ that is called $\alpha$-abelian if, for any $a,b\in R$, and any idempotent $e\in R$, $ea=ae$ and $ab=0$ if and only if $a\alpha (b)=0$ and a ring $R$ is called $\alpha$-semicommutative if, for any $a,b\in R$, $ab=0$ implies $aRb=0$ and $ab=0$ if and only if $a\alpha (b)=0$. According to Krempa (1996), an endomorphism $\alpha$ of a ring $R$ is called rigid if $a\alpha (a)=0$ implies $a=0$ for $a\in R$. We call a ring $R$ $\alpha$-rigid if there exists a rigid endomorphism $\alpha$ of $R$ (see Esmaeili & Hashemi, 2012). There are many ways to generalize the McCoy condition, and we direct reader to the excellent papers (Chen & Tong, 2005; Cui & Chen, 2012; Habibi & Mousavi, 2012; Mousavi, Keshavarz, Rasuli, & Alhevaz, 2011; Song, Liand, & Yang, 2011; Zhao & Liu, 2009) for nice introduction to these topics. Given the fact that $\alpha$-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 $\alpha$-skew M-McCoy left ideal $I$ of $R$ with respect to an endomorphism $\alpha$ of $R$. This notion extends both ideal skew McCoy ideals an $\alpha$-skew McCoy ideals. We do this by considering the $\alpha$-skew McCoy condition on polynomials in $R[M,\alpha ]$ 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. Constructing various examples, we classify how the $\alpha$-skew McCoy ideals relative to a monoid $M$ 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 $\alpha$-skew M-McCoy left ideals.

Definition 2.1 Let $\alpha$ be an endomorphism, a left ideal $I$ of $R$ is called $\alpha$-skew McCoy if the following are satisfied:

1. For any $f(x)=\sum _{i=0}^n{a}_i{x}^i$ and $g(x)=\sum _{j=0}^m{b}_j{x}^j\in R[x,\alpha ],f(x)g(x)\in r_{R[x,\alpha ]}(I[x])$ implies

 i) ideal $I$ of $R$ is a left $\alpha$-skew McCoy if there exists a nonzero element $r\in R$ with $r{b}_j\in r_R(I)$ for all $\mathbb{Q}$,

 ii) ideal $I$ of $R$ is a right $\alpha$-skew McCoy if there exists a nonzero element $s\in R$ with $a_i{\alpha }^i(s)\in r_R(I)$ for all $0\le i\le n$.

2. $ab\in r_R(I)$ if and only if $a\alpha (b)\in r_R(I)$.

If ideal $I$ of $R$ is both left $\alpha$-skew McCoy and right $\alpha$-skew McCoy, then we say that ideal $I$ of $R$ is an $\alpha$-skew McCoy.

Definition 2.2 Let $M$ be a monoid and let $\alpha$ be an endomorphism of a ring $R$, a left ideal $I$ of $R$ is called $\alpha$-skew M-McCoy if the following satisfied:

1. For any $f(x)=\sum _{i=0}^n{a}_i{g}_i$ and $g(x)=\sum _{j=0}^m{b}_j{h}_j\in R[M,\alpha ],f(x)g(x)\in r_{R[M,\alpha ]}(I[M])$ such that $a_i,b_j\in R$ and $g_i,h_j\in M;$

 i) ideal $I$ of $R$ is a left $\alpha$-skew M-McCoy if there exists a nonzero element $r\in R$ with $r{b}_j\in r_R(I)$, for all $0\le j\le m$,

 ii) ideal $I$ of $R$ is a right $\mathbb{Z}$-skew M-McCoy if there exists a nonzero element $s\in R$ with $a_i{\alpha }^i(s)\in r_R(I)$ for all $0\le i\le n$.

2. $ab\in r_R(I)$ if and only if $a\alpha (b)\in r_R(I)$.

If ideal $I$ of $R$ is both left $\alpha$-skew M-McCoy and right $\alpha$-skew M-McCoy, then we say that ideal $I$ of $R$ is an $\alpha$-skew M-McCoy.

Example 2.1 Let $M$ be a monoid and let $\mathbb{Q}$ be an $\alpha$-skew M-McCoy and consider $S=\left\{\left(\begin{array}{ccc}r& a& b\\ 0& r& a\\ 0& 0& r\end{array}\right)\left|r\in \mathbb{Z},a,b\in \mathbb{Q}\right.\right\}$ where $\mathbb{Z}$ and $\mathbb{Q}$ are the sets of all integers and all rational numbers, it is clear that $I=\left\{\left(\begin{array}{ccc}0& a& b\\ 0& 0& a\\ 0& 0& 0\end{array}\right)|a,b\in \mathbb{Q}\right\}$ is the left ideal of $S$. Let $\alpha :S\to S$ be an automorphism defined by $\alpha \left(\left(\begin{array}{ccc}r& a& b\\ 0& r& a\\ 0& 0& r\end{array}\right)\right)=\left(\begin{array}{ccc}r& a/2& b/4\\ 0& r& a/2\\ 0& 0& r\end{array}\right)$. Then

(1) $S$ is not $\alpha$-rigid since $\left(\begin{array}{ccc}0& 0& b\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\alpha \left(\left(\begin{array}{ccc}0& 0& b\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\right)=0$, but $\left(\begin{array}{ccc}0& 0& b\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\ne 0$ if $b\ne 0$.

(2) I of $S$ is $\alpha$-skew M-McCoy left ideal.

let $\alpha =A_0+A_{1}g+\cdots +A_n{g}^n$ and $\beta =B_0+B_{1}g+\cdots +B_m{g}^m\in S[M,\alpha ]-\left\{0\right\}$, where $A_i=$ $\left(\begin{array}{ccc}{a}_{0i}& a_{1i}& a_{2i}\\ 0& a_{0i}& a_{1i}\\ 0& 0& a_{0i}\end{array}\right)$, $B_j=\left(\begin{array}{ccc}{b}_{0i}& b_{1i}& b_{2i}\\ 0& b_{0i}& b_{1i}\\ 0& 0& b_{0i}\end{array}\right)$ for $i=0,\cdots ,n$, $j=0,\cdots ,n$ such that $\alpha \beta \in r_{\mathbb{Q}[M,\alpha ]}(I[M])$. Let $\alpha =\left(\begin{array}{ccc}{\alpha }_0(g)& {\alpha }_1(g)& {\alpha }_2(g)\\ 0& {\alpha }_0(g)& {\alpha }_1(g)\\ 0& 0& {\alpha }_0(g)\end{array}\right)$, $\beta =\left(\begin{array}{ccc}{\beta }_0(g)& {\beta }_1(g)& {\beta }_2(g)\\ 0& {\beta }_0(g)& {\beta }_1(g)\\ 0& 0& {\beta }_0(g)\end{array}\right)$, ${\alpha }_0(g)=a_{00}+a_{01}g+\cdots +a_{0n}{g}^n$, ${\beta }_0(g)=b_{00}+b_{01}g+\cdots +b_{0m}{g}^m$. Since $\alpha \beta \in r_{\mathbb{Q}[M,\alpha ]}(I[M])$ thus for any $h(g)=\left(\begin{array}{ccc}0& 0& \gamma (g)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\in I[M]$, that $\gamma (g)={\gamma }_0+{\gamma }_{1}g+\cdots +{\gamma }_t{g}^t$, $\gamma (g)\alpha \beta =0$, thus $\gamma (g){\alpha }_0(g){\beta }_0(g)=0$. Since $\mathbb{Q}$ is $\alpha$-skew M-McCoy, left ideal, hence ${\gamma }_k{\alpha }^k(a_{0i}{\alpha }^i(b_{0j}))=0$ for all $k=0,...,t$, $i=0,...,n$ and $j=0,...,m$. If set $k=o$, then ${\gamma }_0(a_{0i}{\alpha }^i(b_{0j}))=0$. Since ${\gamma }_0\in R$ is arbitrary, thus $\left(\begin{array}{ccc}0& 0& {\gamma }_0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\in I$. Therefore, $a_{0i}{\alpha }^i(b_{0j})\in r_{\mathbb{Q}}(I)$. If $C=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$. We have $C{b}_j\in r_{\mathbb{Q}}(I)$ for all $j$ and $A_i{\alpha }^i(C)\in r_{\mathbb{Q}}(I)$ for all $i$. Now we consider $\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& a_1& b_1\\ 0& 0& a_1\end{array}\right)$, $\left(\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& a_2& b_2\\ 0& 0& a_2\end{array}\right)\in S$, $\left(\begin{array}{ccc}0& 0& c_3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\in I$, $\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& a_1& b_1\\ 0& 0& a_1\end{array}\right)\left(\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& a_2& b_2\\ 0& 0& a_2\end{array}\right)\in r_{\mathbb{Q}}(I)$. Thus

$\left(\begin{array}{ccc}0& 0& c_3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}{a}_1{a}_2& a_1{b}_2+b_1{a}_2& a_1{c}_2+b_1{b}_2+c_1{a}_2\\ 0& a_1{a}_2& a_1{b}_2+b_1{a}_2\\ 0& 0& a_2\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

and hence $c_3{a}_1{a}_2=0$. Since $\mathbb{Q}$ is $\alpha$-skew M-McCoy, $c_3\alpha (a_2)=0$. Thus

$\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& a_1& b_1\\ 0& 0& a_1\end{array}\right)\left(\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& a_2& b_2\\ 0& 0& a_2\end{array}\right)\in r_{\mathbb{Q}}(I)ifandonlyif$

$\left(\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& a_1& b_1\\ 0& 0& a_1\end{array}\right)\alpha (\left(\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& a_2& b_2\\ 0& 0& a_2\end{array}\right))\in r_{\mathbb{Q}}(I).$

Therefore, $I$ is an $\alpha$-skew M-McCoy left ideal.

One may conjecture that a ring $R$ may be right ideal $\alpha$-skew M-McCoy when $R/I$ and $I$ are both right ideal $\alpha$-skew M-McCoy rings for any nonzero proper ideal $I$ of $R$, where $I$ is considered as a ring without identity. However, the answer is the following:

Example 2.2 Let $F$ be a field and consider $R=U_2(F)$, then $R$ is not right $\alpha$-skew M-McCoy (hence not right ideal $\alpha$-skew M-McCoy) by (Hirano, Hong, & Kim, 2014, Remark 2.2(2)). Note that all nonzero proper ideals of $R$ are

$\left(\begin{array}{cc}F& F\\ 0& 0\end{array}\right),\left(\begin{array}{cc}0& F\\ 0& F\end{array}\right)and\left(\begin{array}{cc}0& F\\ 0& 0\end{array}\right).$

We will show that $R/I$ and $I$ are both right ideal $\alpha$-skew M-McCoy for any nonzero ideal $I$ of $R$. First, let $I=\left(\begin{array}{cc}F& F\\ 0& 0\end{array}\right)$. Then $R/I\cong F$ is right ideal $\alpha$-skew M-McCoy. Let $\gamma \beta \in r_{R[M,\alpha ]}(I[M])$ such that $\gamma =A_0+A_{1}g+\cdots +A_m{g}_m$ and $\beta =B_0+B_{1}h+\cdot \cdot \cdot +B_n{h}_n\notin r_{R[M,\alpha ]}(I[M])$ where $A_i=\left(\begin{array}{cc}{a}_i& b_i\\ 0& 0\end{array}\right)$ and $B_j=\left(\begin{array}{cc}{c}_j& d_j\\ 0& 0\end{array}\right)$ for $1\le i\le m$, $1\le j\le n$. We can write $\gamma =\left(\begin{array}{cc}{\gamma }_1(g)& {\gamma }_2(g)\\ 0& 0\end{array}\right)$ and $\beta =\left(\begin{array}{cc}{\beta }_1(g)& {\beta }_2(g)\\ 0& 0\end{array}\right)$ where ${\gamma }_1(g)={\sum }_{i=0}^m{a}_i{g}_i$, ${\gamma }_2(g)={\sum }_{i=0}^m{b}_i{g}_i$ and ${\beta }_1(h)={\sum }_{j=0}^n{c}_j{h}_j$, ${\beta }_2(h)={\sum }_{j=0}^n{d}_j{h}_j$. From $\gamma \beta \in r_{R[M,\alpha ]}(I[M])$ we have ${\gamma }_1(g){\beta }_1(h)\in r_{R[M]}(I[M])$ and ${\gamma }_1(g){\beta }_2(h)\in r_{R[M]}(I[M])$ if ${\gamma }_1(g)\notin r_{R[M]}(I[M])$, then ${\beta }_1(h)={\beta }_2(h)\in r_{R[M]}(I[M])$ and so $\beta \in r_{R[M,\alpha ]}(I[M])$ a contradiction. Thus ${\gamma }_1(g)\in r_{R[M]}(I[M])$ and ${\gamma }_2(g)\notin r_{R[M]}(I[M])$, i.e., $\gamma =\left(\begin{array}{cc}{\gamma }_1(g)& {\gamma }_2(g)\\ 0& 0\end{array}\right)$. This yields that $\gamma {\alpha }^i(C)\in r_{R[M,\alpha ]}(I[M])$ for every nonzero $C$, in the ideal of $I$ generated by $\text{B,\_j^^,S}$. Next let $I=\left(\begin{array}{cc}0& F\\ 0& F\end{array}\right)$. Then $R/I\cong F$ is right ideal $\alpha$-skew M-McCoy. Let $\gamma \beta \in r_{R[M,\alpha ]}(I[M])$ for $\gamma =A_0+A_{1}g+\cdots +A_m{g}_m=\left(\begin{array}{cc}0& {\gamma }_1(g)\\ 0& {\gamma }_2(g)\end{array}\right)$ and $\beta =B_0+B_{1}h+\cdot \cdot \cdot +B_n{h}_n=$ $\left(\begin{array}{cc}0& {\beta }_1(h)\\ 0& {\beta }_2(h)\end{array}\right)\notin r_{R[M,\alpha ]}(I[M])$, and $\gamma ,\beta \in I[M]$, where $A_i=\left(\begin{array}{cc}0& a_i\\ 0& b_i\end{array}\right)$ and $B_j=\left(\begin{array}{cc}0& c_j\\ 0& d_j\end{array}\right)$ for $1\le i\le m$, $1\le j\le n$, since ${\gamma }_1(g)\notin r_{R[M]}(I[M])$ or ${\gamma }_2(g)\notin r_{R[M]}(I[M])$, we get ${\beta }_2(h)\in r_{R[M]}(I[M])$ from $\gamma \beta \in r_{R[M,\alpha ]}(I[M])$, entailing $\beta =\left(\begin{array}{cc}0& {\beta }_1(h)\\ 0& {\beta }_2(h)\end{array}\right)$. This yields that $\gamma {\alpha }^i(D)\in r_{R[M,\alpha ]}(I[M])$ for every nonzero $D$, in the ideal of $I$ generated by $\text{B,\_j^^,S}$. Finally let $I=\left(\begin{array}{cc}0& F\\ 0& 0\end{array}\right)$. Then $R/I=F\oplus F$ is right $\alpha$-skew M-McCoy. $I$ is clearly right ideal $\alpha$-skew M-McCoy, since $I^2=0$. Similarly we show that $I$ is left ideal $\alpha$-skew M-McCoy. Now we consider $\left(\begin{array}{cc}{F}_1& F_1\\ 0& F_1\end{array}\right)$, $\left(\begin{array}{cc}{F}_2& F_2\\ 0& F_2\end{array}\right)\in R$, $\left(\begin{array}{cc}0& F_3\\ 0& 0\end{array}\right)\in I$, $\left(\begin{array}{cc}{F}_1& F_1\\ 0& F_1\end{array}\right)\left(\begin{array}{cc}{F}_2& F_2\\ 0& F_2\end{array}\right)\in$ $r_{R[M]}(I[M])$. Thus $\left(\begin{array}{cc}0& F_3\\ 0& 0\end{array}\right)\left(\begin{array}{cc}{F}_1{F}_2& 2F_1{F}_2\\ 0& F_1{F}_2\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$, and hence $F_3{F}_1{F}_2=0$, since $I$ is $\alpha$-skew M-McCoy, $F_3{F}_1\alpha (F_2)=0$. Thus $\left(\begin{array}{cc}{F}_1& F_1\\ 0& F_1\end{array}\right)\left(\begin{array}{cc}{F}_2& F_2\\ 0& F_2\end{array}\right)\in r_{R[M]}(I[M])$ if and only if $\left(\begin{array}{cc}{F}_1& F_1\\ 0& F_1\end{array}\right)$ $\alpha \left(\left(\begin{array}{cc}{F}_2& F_2\\ 0& F_2\end{array}\right)\right)\in r_{R[M]}(I[M])$. Therefore, $I$ is an $\alpha$-skew M-McCoy left ideal.

The following example shows that there exists a ring $R$ such that $R$ is $\alpha$-skew M-McCoy. But $R/I$ and $I$ is not $\alpha$-skew M-McCoy.

Example 2.3 Let $\mathbb{Z}$ be the ring of integers and ${\mathbb{Z}}_4$ be the ring of integers modulo 4. Consider a ring

$R=\left\{\left(\begin{array}{cc}a& \bar{b}\\ 0& a\end{array}\right)|a\in \mathbb{Z},\bar{b}\in {\mathbb{Z}}_4\right\}.$

Let $\alpha :R\to R$ be an endomorphism defined by

$\alpha \left(\left(\begin{array}{cc}a& \bar{b}\\ 0& a\end{array}\right)\right)=\left(\begin{array}{cc}a& -\bar{b}\\ 0& a\end{array}\right).$

Then $R$ is $\alpha$-skew M-McCoy by the similar method in the proof of (Hong, Kim, & Kwak, 2003, Proposition 15). However, for an ideal $I=\left\{\left(\begin{array}{cc}a& \bar{0}\\ 0& a\end{array}\right)|a\in 4\mathbb{Z}\right\}$ of $R$, the factor ring

$R/I\cong \left\{\left(\begin{array}{cc}\bar{a}& \bar{b}\\ 0& \bar{a}\end{array}\right)|\bar{a},\bar{b}\in {\mathbb{Z}}_4\right\}$

is not $\alpha$-skew M-McCoy. In fact,

${\left(\left(\begin{array}{cc}\bar{2}& \bar{0}\\ 0& \bar{2}\end{array}\right)+\left(\begin{array}{cc}\bar{2}& \bar{1}\\ 0& \bar{2}\end{array}\right)g\right)}^2=0\in (R/I)[M,\alpha ],$

but

$\left(\begin{array}{cc}\bar{2}& \bar{1}\\ 0& \bar{2}\end{array}\right)\alpha \left(\left(\begin{array}{cc}\bar{2}& \bar{0}\\ 0& \bar{2}\end{array}\right)\right)\ne 0.$

Recall that if $\alpha$ is an endomorphism of a ring $R$, then the map $R[M]\to R[M]$ defined by ${\sum }_{i=0}^n{a}_i{g}_i\to {\sum }_{i=0}^n\alpha (a_i)g_i$ is an endomorphism of the polynomial ring $R[M]$ and clearly this map extends $\alpha$. We shall also denote the extended map $R[M]\to R[M]$ by $\alpha$ and the image of $f\in R[M]$ by $\alpha (f)$.

Theorem 2.1 Let $M$ be a monoid and $\alpha$ be an endomorphism such that ${\alpha }^t=I_R$ for some positive integer $t$. And $I$ be a left ideal of $R$. If $I$ is an $\alpha$-skew M-McCoy, then $I[M,\alpha ]$ is $\alpha$-skew M-McCoy.

Proof. Let $\alpha (y)=f_0+f_{1}y+\cdots +f_n{y}^n$, $\beta (y)=g_0+g_{1}y+\cdots +g_m{g}^m\in R[M,\alpha ][y,\alpha ]-\left\{0\right\}$ with $\alpha (y)\beta (y)\in r_{R[M,\alpha ]}(I[M,\alpha ])$ where $f_i,g_j\in R[M,\alpha ]$ we need to prove each $f_i{\alpha }^i(s)\in r_{R[M]}(I[M])$, and $r{g}_j\in r_{R[M]}(I[M])$ for some nonzero $r,s\in R$.□

Assume that $f_i=a_{i0}+a_{i1}{g}_1+\cdots +a_{i{u}_i}{g}_{u_i},g_i=b_{j0}+b_{j1}{h}_1+\cdots +b_{j{v}_j}{h}_{v_j}$ for each $1\le i\le n$ and $1\le j\le m$, where $a_{i0},...,a_{i{u}_i},b_{j0},...,b_{j{v}_j}\in R$ and $g_i,h_j\in M$. Take a positive integer $k$ such that $k>max\left\{deg(f_i),deg(g_j)\right\}$ for any $1\le i\le n$ and $1\le j\le m$, where the degree is as polynomial in $R[M,\alpha ]$ and the degree of zero polynomial is taken to be 0.

Suppose that $\alpha (g_{tk})=f_0+f_1{g}_{tk+1}+\cdots +f_n{g}_{ntk+n}$, $\beta (h_{tk})=g_0+g_1{h}_{tk+1}+\cdots +g_m{h}_{mtk+m}$. Then $\alpha (g_{tk}),\beta (h_{tk})\in r_{R[M,\alpha ]}(I[M,\alpha ])$. And the set of coefficient of $f{}_i^{,}S$ (resp., $g{}_j^{,}S$) equal the set of coefficients of $\alpha (g_{tk})$ (resp., $\beta (h_{tk})$). It is easy to check that $\alpha (g_{tk})\beta (h_{tk})\in r_{R[M,\alpha ]}(I[M,\alpha ])$, since $\alpha (y)\beta (y)\in r_{R[M,\alpha ]}(I[M,\alpha ])$ and ${\alpha }^{tk}=I_R$. Since $I$ is $\alpha$-skew M-McCoy, there exists $r,s\in R-\left\{0\right\}$ such that $r\beta (h_{tk})\in r_{R[M,\alpha ]}(I[M,\alpha ])$ and $\alpha (g_{tk})s\in r_{R[M,\alpha ]}(I[M,\alpha ])$. And $r\beta (h_{tk})\in r_{R[M,\alpha ]}(I[M,\alpha ])$ implies $r{b}_{jk}\in r_{R[M]}(I[M])$ for any $0\le j\le m$, and $0\le k\le v_j$. Hence $r{g}_j\in r_{R[M]}(I[M])$ for any $0\le j\le n$.

Therefore $I[M,\alpha ]$ is left $\alpha$-skew M-McCoy, $\alpha (g_{tk})s\in r_{R[M,\alpha ]}(I[M,\alpha ])$ implies $a_{il}{\alpha }^{itk+i+l}\in r_{R[M]}(I[M])$ for any $0\le i\le n$ and $0\le l\le u_i$, since ${\alpha }^{i+l}=I_R$. Hence we have

$f_i{\alpha }^i(s)=(a_{i0}+a_{i1}+\cdots +a_{i{u}_i}{g}_{u_i}){\alpha }^i(s)$

$=a_{i0}{\alpha }^{i+0}(s)+a_{i1}{\alpha }^{i+1}(s)g+\cdots +a_{i{u}_i}{\alpha }^{i+u}(s)g_{u_i}\in r_{R[M]}(I[M]),$

for any $0\le i\le n$. Therefore $I[M,\alpha ]$ is right $\alpha$-skew M-McCoy. Now we consider $f,g\in R[M,\alpha ]$, $\lambda \in I[M]$, $fg\in r_{R[M]}(I[M])$. Thus $\lambda fg=0$. Since $I$ is $\alpha$-skew M-McCoy, $\lambda f\alpha (g)=0$. Thus $fg\in r_{R[M]}(I[M])$ if and only if $f\alpha (g)\in r_{R[M]}(I[M])$. Therefore $I[M,\alpha ]$ is $\alpha$-skew M-McCoy. The proof is completed.□

Recall that an element $a$ in $R$ is called regular if $r_R(a)=0=l_R(a)$, i.e., $a$ is not a zero divisor. For subrings of an $\alpha$-skew M-McCoy ring, we have the following.

Proposition 2.1 Let $M$ be a monoid and $\alpha$ be an endomorphism of a ring $R$ and $I$ be an ideal of $R$ satisfying that every nonzero element in $I$ is regular. If $R$ is $\alpha$-skew M-McCoy, then $I$ is $\alpha$-skew M-McCoy (without identity).

Proof. Let $\alpha ={\mathrm{\Sigma }}_{i=0}^n{a}_i{g}_i$, $\beta ={\mathrm{\Sigma }}_{j=0}^m{b}_j{h}_j\in I[M,\alpha ]-\left\{0\right\}$ with $\alpha \beta \in r_{R[M,\alpha ]}(I[M])$, since $I$ is an ideal of $R$ and $R$ is $\alpha$-skew M-McCoy, there exists nonzero elements $r,s\in R$ satisfying $r{b}_j\in r_R(I)$ for any $0\le j\le m$, and $a_i{\alpha }^i(s)\in r_R(I)$ for any $0\le i\le n$. Therefore $tr,st\in I-\left\{0\right\}$ for any nonzero element $t\in I$ (otherwise, if $tr\in r_R(I)$ (resp., $st\in r_R(I)$) for a element $t\in I-\left\{0\right\}$, then $r\in r_R(I)$ (resp., $s\in r_R(I))$. Hence $r=0$ (resp., $s=0$) since every nonzero element in $I$ is regular. This is a contradiction). Consequently, we have

$t(r{b}_j)=(tr)b_j\in r_R(I),(a_i{\alpha }^i(s)){\alpha }^i(t)=a_i{\alpha }^i(st)\in r_R(I)$

for any $0\le i\le n$, and $0\le j\le m$. Thus $I$ is left (resp., right) $\alpha$-skew M-McCoy. Now we consider $a,b\in R$, $c\in I$, $ab\in r_R(I)$. Thus $c(ab)=0$. Since $R$ is $\alpha$-skew M-McCoy, $ca\alpha (b)=0$. Thus $a\alpha (b)\in r_R(I)$. Therefore $I$ is an $\alpha$-skew M-McCoy left ideal.□

Theorem 2.2 Let $M$ be a monoid and $R$ be a ring and $I,J$ be left ideals of $R$. If $I\subseteq J$ and $J/I$ an $\alpha$-skew M-McCoy left ideal of $R/I$,then $J$ is an $\alpha$-skew M-McCoy left ideal of $R$.

Proof. Let $\delta ={\mathrm{\Sigma }}_{i=0}^n{a}_i{g}_i$ and $\beta ={\mathrm{\Sigma }}_{j=0}^m{b}_j{h}_j\in R[M,\alpha ]-\left\{0\right\}$ such that $\delta \beta \in r_{R[M,\alpha ]}(J[M])$. Then ${\mathrm{\Sigma }}_{i=0}^n\bar{a}{g}_i{\mathrm{\Sigma }}_{j=0}^m\bar{b}{h}_j\in r_{R/I[M,\alpha ]}((J/I)[M])$. Thus there exists $r,s$ such that $\bar{a}{\alpha }^i(\bar{s})\in r_{R/I}(J/I)$ and $r\bar{b}\in r_{R/I}(J/I)$, hence $a_i{\alpha }^i(s)\in r_R(J)$ and $r{b}_j\in r_R(J)$.

Now we consider $\bar{a},\bar{b}\in R/I$, $\bar{c}\in J/I$, $\bar{a}\bar{b}\in r_{R/I}(J/I)$. Thus $\bar{c}(\bar{a}\bar{b})=0$. Since $J/I$ is $\alpha$-skew M-McCoy, $\bar{c}\bar{a}\alpha (\bar{b})=0$. Thus $ab\in r_R(J)$ if and only if $a\alpha (b)\in r_R(J)$. Therefore, $J$ is an $\alpha$-skew M-McCoy.□

The following is an immediate corollary of Theorem 2.2:

Corollary 2.1 Let $R$ be a ring and $I$ an left ideal of $R$. If $R/I$ is $\alpha$-skew M-McCoy then $R$ is an $\alpha$-skew M-McCoy ring.

To prove Proposition 2.4, we state the following propositions:

Proposition 2.2 If $I$ is an $\alpha$-skew M-McCoy left ideal of $R$ and for some $a,b,c\in R$ and some integer $n\ge 1$, $ab\in r_R(I)$ and $a{c}^n{\alpha }^n(b)\in r_R(I)$ then $acb\in r_R(I)$.

Proof. Consider $\gamma =a(1-cg)$, $\beta =(1+cg+\cdots +c^{n-1}{g}^{n-1})b\in R[M,\alpha ]$, $\gamma \beta =ab-a{c}^n{\alpha }^n(b)g^n\in$ $r_{R[M,\alpha ]}(I(M))$. Since $I$ is an $\alpha$-skew M-McCoy left ideal of $R$, so $acb\in r_R(I)$.□

Recall that a left ideal $I$ of $R$ is called $\alpha$-abelian if, for any $a,b\in R$ and any idempotent $e\in R$, we have the following conditions:

1. $ea-ae\in r_R(I)$,

2. $ab\in r_R(I)$ if and only if $a\alpha (b)\in r_R(I)$.

Next, we show that every $\alpha$-skew M-McCoy left ideal of $R$ is an $\alpha$-abelian left ideal.

Proposition 2.3 If $I$ is an $\alpha$-skew M-McCoy left ideal of $R$, then $I$ is an $\alpha$-abelian left ideal.

Proof. Assume that $I$ is an $\alpha$-skew M-McCoy left ideal of $R$. Consider $e=e^2\in R$ and let $a=e$, $b=(1-e)$, $c=er(1-e)$ with $r\in R$. Then clearly $ab\in r_R(I)$ and $e^2=0\in r_R(I)$ and hence $a{c}^2{\alpha }^2(b)\in r_R(I)$ and then by Proposition 2.3, $acb\in r_R(I)$. So $er-ere\in r_R(I)$. Let $a_1=1-e$, $b_1=e$, and $c_1=(1-e)re$, we also have $a_1{b}_1{c}_1\in r_R(I)$. So $re-ere\in r_R(I)$ then $re-er\in r_R(I)$.□

Recall that a left ideal $I$ of $R$ is called $\alpha$-semicommutative if, for any $a,b\in R$ we have the following conditions:

1. $ab\in r_R(I)$ then $aRb\subseteq r_R(I)$,

2. $ab\in r_R(I)$ if and only if $a\alpha (b)\in r_R(I)$.

A ring $R$ is called locally finite if every finite subset of $R$ 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 $R$ be a locally finite ring and $I$ be an $\alpha$-skew M-McCoy left ideal of $R$. Then $I$ is $\alpha$-semicommutative left ideal of $R$.

Proof. Let $ab\in r_R(I)$ with $a,b\in R$. For any $r\in R$, since $R$ is locally finite there exists integers $m,k\ge 1$ such that $r^m=r^{m+k}$. So we obtain inductively $r^m=r^m{r}^k=r^m{r}^{2k}=\cdots =r^m{r}^{mk}=r^{m(k+1)}$, put $h=k+1$ then $r^m=(r^m{)}^h$ with $h\ge 2$. Notice that $r^{(h-1)m}=r^{(h-2)m}{r}^m=r^{(h-2)m}(r^m{)}^h=r^{2(h-1)m}=(r^{(h-1)m}{)}^2$. Hence, $r^{(h-1)m}$ is an idempotent and so by Proposition 2.3, $a{r}^{(h-1)m}-r^{(h-1)m}a\in r_R(I)$ and $ab{r}^{(h-1)m}-r^{(h-1)m}ab\in r_R(I)$. Thus, $ab{r}^{(h-1)m}\in r_R(I)$. On the other hand by Proposition 2.3, $a{r}^{(h-1)m}-r^{(h-1)m}a\in r_R(I)$, so $a{r}^{(h-1)m}b-r^{(h-1)m}ab\in r_R(I)$, and hence $a{r}^{(h-1)m}b\in r_R(I)$. Since $I$ is an $\alpha$-skew M-McCoy left ideal of $R$ so $a{r}^{(h-1)m}{\alpha }^{(h-1)m}(b)\in r_R(I)$, and by Proposition 2.2, we imply that $arb\in r_R(I)$ for all $r\in R$.□

Let $\alpha$ be an endomorphism of a ring $R$ and $M_n(R)$ be the $n\times n$ matrix over ring $R$ and $\bar{\alpha }:M_n(R)\to M_n(R)$ defined by $\bar{\alpha }(a_{ij})=(\alpha (a_{ij}))$. Then $\bar{\alpha }$ is an endomorphism of $M_n(R)$. It is obvious that, the restriction of $\bar{\alpha }$ to $D_n(R)$ is an endomorphism of $D_n(R)$, where $D_n(R)$ is the $n\times n$ diagonal matrix ring over $R$. We also denote $\bar{\alpha }{|}_{D_n(R)}$ by $\bar{\alpha }$.

Proposition 2.5 Let $\alpha$ be an endomorphism of a ring $R$. Then $D_n(I)$ is an $\bar{\alpha }$-skew M-McCoy left ideal of $D_n(R)$ if $I$ is an $\alpha$-skew M-McCoy left ideal for any $n$.

Proof. Let $\alpha =A_0+A_1{g}_1+\cdots +A_p{g}_p$ and $\beta =B_0+B_1{h}_1+\cdots +B_q{h}_q\in D_n(R)[M,\bar{\alpha }]$ satisfying $\alpha \beta \in r_{D_n(R)[M,\bar{\alpha }]}(D_n(I)[M])$, where

$A_i=\left(\begin{array}{cccc}{a}_{11}^{(i)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}\end{array}\right)$ and $B_j=\left(\begin{array}{cccc}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& b_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& b_{nn}^{(j)}\end{array}\right)$.

Then from $\alpha \beta \in r_{D_n(R)[M,\bar{\alpha }]}(D_n(I)[M])$, it follows that,

$\left(\sum _{i=0}^p{a}_{ss}^{(i)}{g}_i\right)\left(\sum _{j=0}^q{b}_{ss}^{(j)}{h}_j\right)\in r_{R[M,\alpha ]}(I[M])$

for each $1\le s\le n$. Since $I$ is an $\alpha$-skew M-McCoy left ideal of $R$, then there exists $r,s\in R$ such that $r(b_{ss}^{(j)})\in r_R(I)$ and $a_{ss}^{(i)}{\alpha }^i(s)\in r_R(I)$ for any $1\le i\le p$ and $1\le j\le q$. Therefore

$r(B_j)=\left(\begin{array}{cccc}r(b_{11}^{(j)})& 0& \cdots & 0\\ 0& r(b_{22}^{(j)})& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& r(b_{nn}^{(j)})\end{array}\right)\in r_{D_n(R)}(D_n(I))and$

$A_i{\alpha }^i(B_j)=\left(\begin{array}{cccc}{a}_{11}^{(i)}{\alpha }^i(b_{11}^{(j)})& 0& \cdots & 0\\ 0& a_{22}^{(i)}{\alpha }^i(b_{22}^{(j)})& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& 0& \cdots & a_{nn}^{(i)}{\alpha }^i(b_{nn}^{(j)})\end{array}\right)\in r_{D_n(R)}(D_n(I)).$

Now we consider $A_i=\left(\begin{array}{cccc}{a}_{11}^{(i)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}\end{array}\right)$, $B_j=\left(\begin{array}{cccc}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& b_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& b_{nn}^{(j)}\end{array}\right)\in D_n(R),$ $C_k=$ $\left(\begin{array}{cccc}{c}_{11}^{(k)}& 0& \cdots & 0\\ 0& c_{22}^{(k)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& c_{nn}^{(k)}\end{array}\right)\in D_n(I)$,

$\left(\begin{array}{cccc}{a}_{11}^{(i)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}\end{array}\right)\left(\begin{array}{cccc}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& b_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& b_{nn}^{(j)}\end{array}\right)\in r_{D_n(R)}(D_n(I)).$

Thus $\left(\begin{array}{cccc}{c}_{11}^{(k)}& 0& \cdots & 0\\ 0& c_{22}^{(k)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& c_{nn}^{(k)}\end{array}\right)\left(\begin{array}{cccc}{a}_{11}^{(i)}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}{b}_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}{b}_{nn}^{(j)}\end{array}\right)=\left(\begin{array}{cccc}0& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& 0\end{array}\right)$, and hence $c_{11}^{(k)}{a}_{11}^{(i)}{b}_{11}^{(j)}=0$. Since $I$ is an $\alpha$-skew M-McCoy, $c_{11}^{(k)}{a}_{11}^{(i)}\alpha (b_{11}^{(j)})=0$. Thus $\left(\begin{array}{cccc}{a}_{11}^{(i)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}\end{array}\right)\left(\begin{array}{cccc}{a}_{11}^{(i)}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}{b}_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}{b}_{nn}^{(j)}\end{array}\right)\in r_{D_n(R)}(D_n(I))$ if and only if $\left(\begin{array}{cccc}{a}_{11}^{(i)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}\end{array}\right)\left(\begin{array}{cccc}{a}_{11}^{(i)}{b}_{11}^{(j)}& 0& \cdots & 0\\ 0& a_{22}^{(i)}{b}_{22}^{(j)}& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& a_{nn}^{(i)}{b}_{nn}^{(j)}\end{array}\right)\in r_{D_n(R)}(D_n(I))$.

Therefore $D_n(I)$ is an $\bar{\alpha }$-skew M-McCoy left ideal.□

Additional information

Funding

The authors received no direct funding for this research.

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.