On nil-McCoy rings relative to a monoid

Abstract

The concept of nil-M-McCoy (nil-McCoy ring relative to monoid M), which are generalizations of McCoy ring and nil-M-Armendariz rings have been introduced, and we investigate their properties. It is shown that every NI ring is nil-M-McCoy for any unique product monoid M, it has also been shown that every semicommutative rings is nil-M-McCoy for any unique product monoid and any strictly totally ordered monoid M. Moreover, it is proved that for an ideal I of R, if I is semicommutative and R / I is nil-M-McCoy then R is nil-M-McCoy for any strictly totally ordered monoid. We extend and unify many known results related to McCoy rings and nil-Armendariz ring.

Keywords:

• M-McCoy rings
• Nil-M-Armendariz rings
• Nil-M-McCoy rings
• u.p.-monoid

AMS Subject Classifications:

• 16U99
• 16S15

Public Interest Statement

The current paper attempts to generalize Nil-macCoy rings with Monoid to present a novel work with the title of Nil-M-MacCoy. The survey of literature demonstrates that studies on Nill-Armendariz and Nil-MacCoy and Nil-M-Armendariz and have already been researched. The theorems, lemmas and corollaries which have been already studied concerning this topic to be compared with the new work to generalize or complete it. It paves the way for the willing researchers to continue the trend with the novel areas that have been investigated.

1. Introduction

All rings considered here are associative with identity. For a ring R, nil(R) denotes the set of nilpotent elements. Rege and Chhawchharia (1997) introduced the notion of an Armendariz ring Recall that a ring R is called reduced if $a^2=0$ implies that $a=0$, for all $a\in R$; R is symmetric if $abc=0$ implies $acb=0$, for $a,b,c\in R$; R is reversible if $ab=0$ implies $ba=0$, for all $a,b\in R$; R is semicommutative if $ab=0$ implies $aRb=0$, for all $a,b\in R$.

Recall that a monoid M is called an u.p.-monoid (unique product monoid) if for any two nonempty finite subsets $A,B\subseteq M$ there exists an element $g\in M$ that can be written uniquely in the form ab where $a\in A$ and $b\in B$. The class of u.p.monoids is quite large and important (see Birkenmeier & Park, 2003).

According to Nielsen (2006), a ring R is called right McCoy whenever polynomials $f(x),g(x)\in R[x]-\{0\}$ satisfy $f(x)g(x)=0$, there exists a nonzero $r\in R$ such that $f(x)r=0$. We define left McCoy ring similarly. If a ring is both left and right McCoy, we say that the ring is a McCoy ring. Liu (2005), studied a generalization of Armendariz ring, which is called M-Armendariz ring if whenever $\alpha ={\sum }_{i=1}^n{a}_i{g}_i$, $\beta ={\sum }_{j=1}^m{b}_j{h}_j\in R[M]$, with $g_i,h_j\in M$ satisfy $\alpha \beta =0$, then $a_j{b}_j=0$ for each $i,j$.

Zhao, Zhu, and Gu (2012) introduced the notion of a nil-McCoy ring. A ring R is said to be right nil-McCoy if $f(x)g(x)\in nil(R)[x]$, where $f(x)={\sum }_{i=0}^m{a}_i{x}^i$, $g(x)={\sum }_{j=0}^n{b}_j{x}^j\in R[x]-\{0\}$, implies that there exists $r\in R-\{0\}$ such that $a_{i}r\in nil(R)$ for $0\le i\le m$. Left nil-McCoy ring are defined similarly. A ring is nil-McCoy if it is both left and right nil-McCoy. According to Hashemi (2010), a ring R is called right M-McCoy if whenever $\alpha ={\sum }_{i=1}^n{a}_i{g}_i$, $\beta ={\sum }_{j=1}^m{b}_j{h}_j\in R[M]-\{0\}$, with $g_i,h_j\in M$, $a_i,b_j\in R$ satisfy $\alpha \beta =0$, then $\alpha r=0$ for some nonzero $r\in R$. Left M-McCoy ring are defined similarly. Also, if R is both left and right M-McCoy, then we say R is M-McCoy.

Our results also generalized and unify the above-mentioned concepts by introducing the notion of nil-M-McCoy ring.

2. Nil-McCoy rings relative to a monoid

We begin this section by the following definition and also we study properties of nil-M-McCoy rings.

Definition 2.1

Let M be a monoid and R a ring. We say that R is right nil-M-McCoy (right nil-McCoy ring relative to M), if whenever $\alpha =a_1{g}_1+\cdots +a_n{g}_n$, $\beta =b_1{h}_1+\cdots +b_m{h}_m\in R[M]-\{0\}$, 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 we say R is nil-M-McCoy.

Let $M=(\mathcal{N}\cup \{0\},+)$. Then a ring R is nil-M-McCoy (resp., nil-M-Armendariz) if and only if R is nil-McCoy (resp., nil-Armendariz). If $M=\{e\}$, then for any ring R, R is nil-M-McCoy and nil-M-Armendariz.

Example 2.1

Let ${\mathcal{Z}}_2$ be the ring of integers modulo 2 and R be a nil-Armendariz ring and$$R={\mathcal{Z}}_2⟨a,b,c,d,e⟩/I,$$

where I is the ideal generated by the relations: $ac=0,ad+bc=0,bd=0,ea=eb=ec=ed=ee=de=ce=be=0$. Get S be subring generated by elements not involving e. Let $\alpha =a+bg$ and $\beta =c+dh$ such that $g,h\in R[M]-\{0\}$ and let $g=h$ then $\alpha \beta =0$. It is straightforward to see that the element as is not nilpotent for any nonzero $s\in S$. However, R is nil-M-McCoy as $\alpha e,e\beta \in nil(R)[M]$ for any elements $\alpha ,\beta \in R[M]$.

For any $\alpha \in R[M]$, we denote by $C_{\alpha }$ the set of all coefficients of $\alpha$.

Proposition 2.1

For any "u.p."-monoid M, every NI ring is nil-M-McCoy.

Proof

Let M be an "u.p."-monoid and R be a NI ring. Suppose that $\alpha =a_1{g}_1+\cdots +a_m{g}_m,\beta =b_1{h}_1+\cdots +b_n{h}_n\in R[M]-\{0\}$ satisfy $\alpha \beta \in nil(R)[M]$. Since nil(R) is an ideal of R, R / nil(R) is M-McCoy by Liu (2005, Proposition 2.6). For any $\alpha ={\sum }_{i=1}^m{a}_i{g}_i\in R[M]$, denote $\overline{\alpha }={\sum }_{i=1}^m(a_i+nil(R))g_i\in (R/nil(R))[M]$. It is easy to see that the mapping $\phi :R[M]\to (R/nil(R))[M]$. Defined by $\phi ({\alpha }_i)={\overline{\alpha }}_i$ is a ring homomorphism. Since $\alpha \beta \in nil(R)[M]$, $C_{\alpha \beta }\subseteq nil(R)$. Hence, we have $\overline{\alpha }\overline{\beta }=\overline{\alpha \beta }\in nil(\overline{R})[M]$. So ${\overline{a}}_i{\overline{b}}_j\in nil(\overline{R})=0$ for i, j, since R / nil(R) is M-McCoy. Thus, $a_i{b}_j\in nil(R)$ for i, j. Choosing $r=b_n\ne 0$ and $s=a_m\ne 0$, we have $a_{i}r\in nil(R)$ and $s{b}_j\in nil(R)$, for i, j. Therefore, R is nil-M-McCoy.

Example 2.2

Let M be "u.p."-monoid and K be field. Suppose $S=K<a|a^2=0>$. Then the ring $R=\left(\begin{array}{cc}s& \mathit{as}\\ as& s\end{array}\right)$ is nil-M-McCoy, since R is a NI ring.

To prove Theorem 2.1, we state the following lemmas.

Lemma 2.1

Let M be cyclic group of order $n\ge 2$ and R a ring with $1\ne 0$. Then R is not nil-M-McCoy.

Proof

Suppose that $M=\{e,h,\dots ,h^{n-1}\}$. Let $\alpha =1e+1h+\cdots +1h^{n-1}$ and $\beta =1e+(-1)h$ then $\alpha \beta \in nil(R)[M]$, but $a_{i}c=1c\notin nil(R)$ for each nonzero $c\in R$. Therefore, R is not right nil-M-McCoy. Similarly R is not left nil-M-McCoy. Therefore, R is not nil-M-McCoy.

Lemma 2.2

Let M be a monoid and N a submonoid of M. If R is nil-M-McCoy, then R is nil-N-McCoy.

Proof

It is clear by Liu (2005, lemma 1.12) and Hashemi (2010, lemma 1.12).

Lemma 2.3

Liu (2005, Lemma 1.13). Let M and N be "u.p."-monoid. Then so is the monoid $M\times N$.

Example 2.3

Let $M,N\in M_2(\mathcal{R})$ and let M, N be “u.p.”-monoids. Let$$A=\left\{\left(\left(\begin{array}{cc}{a}_i& b_i\\ c_i& d_i\end{array}\right),\left(\begin{array}{cc}{e}_i& f_i\\ g_i& h_i\end{array}\right)\right)\in M\times N\mid i=1,2,...,p\right\}$$

and$$B=\left\{\left(\left(\begin{array}{cc}{a}_j^{\prime }& b_j^{\prime }\\ c_j^{\prime }& d_j^{\prime }\end{array}\right),\left(\begin{array}{cc}{e}_j^{\prime }& f_j^{\prime }\\ g_j^{\prime }& h_j^{\prime }\end{array}\right)\right)\in M\times N\mid j=1,2,\dots ,q\right\}$$. Since M is a "u.p."-monoid, there exist i, j with $1\le i\le p$ and $1\le j\le q$ such that $\left(\begin{array}{cc}{a}_i& b_i\\ c_i& d_i\end{array}\right)\left(\begin{array}{cc}{a}_j^{\prime }& b_j^{\prime }\\ c_j^{\prime }& d_j^{\prime }\end{array}\right)$ is uniquely presented by considering two subsets$$\left\{\left(\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right),...,\left(\begin{array}{cc}{a}_p& b_p\\ c_p& d_p\end{array}\right)\right\}and\left\{\left(\begin{array}{cc}{a}_1^{\prime }& b_1^{\prime }\\ c_1^{\prime }& d_1^{\prime }\end{array}\right),...,\left(\begin{array}{cc}{a}_q^{\prime }& b_q^{\prime }\\ c_q^{\prime }& d_q^{\prime }\end{array}\right)\right\}ofM.$$

Consider two subsets$$\left\{\left(\begin{array}{cc}{e}_k& f_k\\ g_k& h_k\end{array}\right)\in N\mid \left(\left(\begin{array}{cc}{a}_i& b_i\\ c_i& d_i\end{array}\right),\left(\begin{array}{cc}{e}_k& f_k\\ g_k& h_k\end{array}\right)\right)\in A\right\}and$$$$\left\{\left(\begin{array}{cc}{e}_l^{\prime }& f_l^{\prime }\\ g_l^{\prime }& h_l^{\prime }\end{array}\right)\in N\mid \left(\left(\begin{array}{cc}{a}_j^{\prime }& b_j^{\prime }\\ c_j^{\prime }& d_j^{\prime }\end{array}\right),\left(\begin{array}{cc}{e}_l^{\prime }& f_l^{\prime }\\ g_l^{\prime }& h_l^{\prime }\end{array}\right)\right)\in B\right\}\text{of}N.$$

Since N is a “u.p.”-monoid, there exist $k,l$ such that $\left(\begin{array}{cc}{e}_k& f_k\\ g_k& h_k\end{array}\right)\left(\begin{array}{cc}{e}_l^{\prime }& f_l^{\prime }\\ g_l^{\prime }& h_l^{\prime }\end{array}\right)$ is uniquely presented.

Now it is easy to see that $\left(\left(\begin{array}{cc}{a}_i& b_i\\ c_i& d_i\end{array}\right),\left(\begin{array}{cc}{e}_k& f_k\\ g_k& h_k\end{array}\right)\right)\left(\left(\begin{array}{cc}{a}_j^{\prime }& b_j^{\prime }\\ c_j^{\prime }& d_j^{\prime }\end{array}\right),\left(\begin{array}{cc}{e}_l^{\prime }& f_l^{\prime }\\ g_l^{\prime }& h_l^{\prime }\end{array}\right)\right)$ is uniquely presented. Thus $M\times N$ is a “u.p.”-monoid.

Recall that a monoid M is called torsion-free if the following property holds: if $g,h\in M$ and $k\ge 1$ are such that $g^k=h^k$, then $g=h$.

Let T(H) be the set of elements of finite order in an abelian group H. H is said to be torsien-free If T(H)={e}.

Theorem 2.1

Let H be finitely generated abelian group. Then the following conditions on H are equivalent:

(1)	H is torsion-free.
(2)	There exists a ring R with $|R|\ge 2$ such that nil-H-McCoy.

Proof

$(1)\Rightarrow (2)$ If H is finitely generated abelian group with $T(H)=\{e\}$, then $H\approx \mathcal{Z}\times \mathcal{Z}\times \dots \times \mathcal{Z}$, a finite direct product of the group $\mathcal{Z}$, by Lemma 2.3, H is “u.p.”-monoid. Let R be a NI ring. Then, by Proposition 2.1, R is nil-H-McCoy.

$(2)\Rightarrow (1)$ If $h\in T(H)$ and $h\ne e$, then $N=⟨h⟩$ is a cyclic group of finite order. If a ring $R\ne \{0\}$ is nil-H-McCoy, then by Lemma 2.2, R is nil-N-McCoy, a contradiction with Lemma 2.1. Therefore, every ring $R\ne \{0\}$ is not nil-H-McCoy.

A ring R is called right Ore if given $a,b\in R$ with b regular, there exist $a_1,b_1\in R$ with $b_1$ regular such that $a{b}_1=b{a}_1$. left Ore rings can be defined similarly.

Proposition 2.2

Let M be a monoid and R a right Ore ring with its classical right quotient ring Q. If R is right nil-M-McCoy, then Q is right nil-M-McCoy.

Proof

Let $\alpha ={\sum }_{i=0}^m{\alpha }_i{g}_i$, $\beta ={\sum }_{j=0}^n{\beta }_j{h}_j$ be nonzero polynomials of Q[M] such that $\alpha \beta \in nil(Q)[M]$. Since Q is a classical right quotient ring, we assume that ${\alpha }_i=a_i{u}^{-1}$ and ${\beta }_j=b_j{v}^{-1}$ for $a_i,b_j\in R$ for all i, j and regular elements $u,v\in R$. For each j, there exists $c_j\in R$ and a regular element $w\in R$ such that $u^{-1}{b}_j=c_j{w}^{-1}$. Denote ${\alpha }^{\prime }={\sum }_{i=0}^m{a}_i{g}_i$ and ${\beta }^{\prime }={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0\}$. We have $\alpha \beta ={\sum }_{i=0}^m{\sum }_{j=0}^n{\alpha }_i{\beta }_j{g}_i{h}_j={\sum }_{i=0}^m{\sum }_{j=0}^n{a}_i{u}^{-1}{b}_j{v}^{-1}{g}_i{h}_j={\sum }_{i=0}^m{\sum }_{j=0}^n{a}_i{c}_j{w}^{-1}{v}^{-1}{g}_i{h}_j={\sum }_{i=0}^m{\sum }_{j=0}^n{a}_i{c}_j{(vw)}^{-1}{g}_i{h}_j={\alpha }^{\prime }{\beta }^{\prime }{(vw)}^{-1}$. So ${\alpha }^{\prime }{\beta }^{\prime }\in nil(R)[M]$. Since R is right nil-M-McCoy, there exists a nonzero element $c\in R$ such that $a_{i}c\in nil(R)$. So $a_{i}c={\alpha }_i(uc)\in nil(R)$, for each i, and uc is a nonzero element in Q. Hence, Q is a right nil-M-McCoy ring.

Theorem 2.2

Let M be a monoid with $|M|\ge 2$. Then the finite direct sum of nil-M-McCoy rings is nil-M-McCoy.

Proof

It suffices to show that if $R_1,R_2$ are nil-M-McCoy rings, then so is $R_1\oplus R_2$. Let $\alpha =(a_1^1,b_1^1)g_1+\cdots +(a_m^1,b_m^1)g_m$ and $\beta =(a_1^2,b_1^2)h_1+\cdots +(a_n^2,b_n^2)h_n\in (R_1\oplus R_2)[M]-\{0\}$ be such that $\alpha \beta \in nil(R_1\oplus R_2)[M]$. Therefore we show that there exists $r,s\in R_1\oplus R_2$ such that $a_{i}r\in nil(R_1\oplus R_2)$ and $s{b}_j\in nil(R_1\oplus R_2)$ for each $a_i=(a_i^1,b_i^1)$, $b_j=(a_j^2,b_j^2)$ and for all $r=(r_1,r_2),s=(s_1,s_2)$. Now let $f_1=a_1^1{g}_1+\cdots +a_m^1{g}_m$, $g_1=b_1^1{g}_1+\cdots +b_m^1{g}_m$, $f_2=a_1^2{h}_1+\cdots +a_n^2{h}_n$, $g_2=b_1^2{h}_1+\cdots +b_n^2{h}_n$. Then $f_1{f}_2\in nil(R_1)[M]$ and $g_1{g}_2\in nil(R_2)[M]$. Since by hyopothesis $R_1$ and $R_2$ are nil-M-McCoy $a_i^1{r}_1\in nil(R_1)$, $b_j^1{r}_2\in nil(R_2)$ for each $1\le i\le m$. Thus for each $1\le i\le m$$(a_i^1,b_i^1)(r_1,r_2)\in nil(R_1\oplus R_2)$, and also $s_1{a}_j^2\in nil(R_1)$, $s_2{b}_j^2\in nil(R_2)$ for each $1\le j\le n$. Thus for each $1\le j\le n$$(s_1,s_2)(a_j^2,b_j^2)\in nil(R_1\oplus R_2)$. Therefore, $R_1\oplus R_2$ is nil-M-McCoy.

Example 2.4

Let R be an M-McCoy reduced ring, and M a monoid with $|M|\ge 2$ and let$$T_n(R)=\{\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ 0& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_{\mathit{nn}}\end{array}\right)|a_{\mathit{ij}}\in R\}.$$

Then $T_n(R)$ is not M-McCoy for $n\ge 4$ by Ying et al. (2008, Theorem 2.1). But $T_n(R)$ is nil-M-McCoy by Theorem 2.3 below. Hence, a nil-M-McCoy ring is not a trivial extension of an M-McCoy.

Theorem 2.3

Let M be a monoid with $|M|\ge 2$. Then the following conditions are equivalent:

(1)	R is nil-M-McCoy.
(2)	$T_n(R)$ is nil-M-McCoy.

Proof

$(1)\Rightarrow (2)$ In a similar way, proposition 2.12 (Nikmehr, Fatahi, & Amraei, 2011). It is easy to see that there exists an isomorphism of rings $T_n(R)[M]\to T_n(R[M])$ defined$$\sum _{i=1}^n\left(\begin{array}{cccc}{a}_{11}^i& a_{12}^i& \cdots & a_{1n}^i\\ 0& a_{22}^i& \cdots & a_{2n}^i\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_{\mathit{nn}}^i\end{array}\right)g_i\to \left(\begin{array}{cccc}{\sum }_{i=1}^n{a}_{11}^i{g}_i& {\sum }_{i=1}^n{a}_{12}^i{g}_i& \cdots & {\sum }_{i=1}^n{a}_{1n}^i{g}_i\\ 0& {\sum }_{i=1}^n{a}_{22}^i{g}_i& \cdots & {\sum }_{i=1}^n{a}_{2n}^i{g}_i\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\sum }_{i=1}^n{a}_{\mathit{nn}}^i{g}_i\end{array}\right).$$

Suppose that $\alpha =A_1{g}_1+\cdots +A_n{g}_n$ and $\beta =B_1{h}_1+\cdots +B_m{h}_m\in T_n(R)[M]-\{0\}$ are such that $\alpha \beta \in nil(T_n(R))[M]$, where $A_i,B_j\in T_n(R)$. We claim there exists $B\in T_n(R)$ such that $A_{i}B\in nil(T_n(R))$ for each i. Assume that$$A_i=\left(\begin{array}{cccc}{a}_{11}^i& a_{12}^i& \cdots & a_{1n}^i\\ 0& a_{22}^i& \cdots & a_{2n}^i\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_{\mathit{nn}}^i\end{array}\right),B=\left(\begin{array}{cccc}{c}_{11}^j& c_{12}^j& \cdots & c_{1n}^j\\ 0& c_{22}^j& \cdots & c_{2n}^j\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & c_{\mathit{nn}}^j\end{array}\right),$$

and let ${\alpha }_s={\sum }_{i=1}^n{a}_{\mathit{ss}}^i{g}_i\in R[M]$ and ${\beta }_s={\sum }_{j=1}^m{c}_{\mathit{ss}}^j{h}_j\in R$. Also by observing that$$nil(T_n(R))=\left(\begin{array}{cccc}nil(R)& R& \cdots & R\\ 0& nil(R)& \cdots & R\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & nil(R)\end{array}\right),$$

thus, we have ${\alpha }_s{\beta }_s\in nil(R)[M]$ for each $1\le s\le n$. Since R is a nil-M-McCoy ring, there exists some positive integer $m_{i_s}$ such that ${(a_{\mathit{ss}}^i{b}_{\mathit{ss}}^j)}^{m_{i_s}}=0$ for any s and any i. Let $m_i=max\{m_{i_s}|1\le s\le n\}$. Then ${({(A_{i}B)}^{m_i})}^n=0$ and so $A_{i}B\in nil(T_n(R))$. Therefore, $T_n(R)$ is nil-M-McCoy.

$(2)\Rightarrow (1)$ Suppose that $T_n(R)$ is nil-M-McCoy. Note that R is isomorphic to the subring$$\left\{\left(\begin{array}{cccc}a& 0& \cdots & 0\\ 0& a& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a\end{array}\right)|a\in R\right\}$$

of $T_n(R)$. Thus, R is nil-M-McCoy since each sub ring of a nil-M-McCoy ring is also nil-M-McCoy.

Let R be a ring and let$$S_n(R)=\left\{\left(\begin{array}{cccc}a& a_{12}& \cdots & a_{1n}\\ 0& a& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a\end{array}\right)|a,a_{\mathit{ij}}\in R\right\},$$

and$$T(R,n)=\{\left(\begin{array}{cccc}{a}_1& a_2& \cdots & a_n\\ 0& a_1& \cdots & a_{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & a_1\end{array}\right)|a_{\mathit{ij}}\in R\},$$

and T(R, R) be the trivial extension of R by R. Using the same method in the proof of Theorem 2.3, we have the following results:

Corollary 2.1

Let M be a monoid with $|M|\ge 2$. Then the following conditions are equivalent:

(1)	R is nil-M-McCoy;
(2)	$S_n(R)$ is nil-M-McCoy;
(3)	T(R, n) is nil-M-McCoy;
(4)	T(R, R) is nil-M-McCoy;
(5)	$R[x]/⟨x^n⟩$ is nil-M-McCoy for each $n\ge 2$.

Proof

Using the same method in the proof of Theorem 2.3 we have $(1)\iff (2)$, $(1)\iff (3)$ and $(1)\iff (4)$. It is easy to see that T(R, n) is a sub ring of the triangular matrix rings, with matrix addition and multiplication. We can denote elements of T(R, n) by $(a_0,a_1,\dots ,a_{n-1})$, then T(R, n) is a ring with addition point-wise and multiplication given by$$(a_0,a_1,\dots ,a_{n-1})(b_0,b_1,\dots ,b_{n-1})=(a_0{b}_0,a_0{b}_1+a_1{b}_0,\dots ,a_0{b}_{n-1}+\dots +a_{n-1}{b}_0),$$

for each $a_i,b_j\in R.$ On the other hand, there is a ring isomorphism $\phi :\frac{R[x]}{<x^n>}\to T(R,n)$, since by, $\phi (a_0+a_{1}x+\dots +a_{n-1}{x}^{n-1})=(a_0,a_1,\dots ,a_{n-1})$, with $a_i\in R$, $0\le i\le n-1$. So $T(R,n)\approx \frac{R[x]}{<x^n>}$, where R[x] is the ring of polynomial in an indeterminate x and $<x^n>$ is the ideal generated by $x^n$. Therefore $(3)\iff (5)$.Thus by similarly method we have $(4)\iff (5)$. Therefore all conditions are equivalent.

Let R be a ring and M a monoid. Let $G_3(R)=\{\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)|a_{\mathit{ij}}ϵR\}$. The $G_3(R)$ is a subring of $3\times 3$ full matrix ring $M_3(R)$ under usual addition and multiplication. In fact, $G_3(R)$ possesses the similar form of both the ring of all lower triangular matrices and the ring of all upper triangular matrices. A natural problem asks if the nil-M-McCoy property of such subring of $M_n(R)$ coincides with that of R. This inspires us to consider the nil-M-McCoy property of $G_3(R)$.

Theorem 2.4

Let M be a monoid with $|M|\ge 2$. Then the following conditions are equivalent:

(1)	R is nil-M-McCoy;
(2)	$G_3(R)$ is nil-M-McCoy.

Proof

$(1)\Rightarrow (2)$ We first show that $nil(G_3(R))=\left(\begin{array}{ccc}nil(R)& 0& 0\\ R& nil(R)& R\\ 0& 0& nil(R)\end{array}\right).$ Suppose that $\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)\in \left(\begin{array}{ccc}nil(R)& 0& 0\\ R& nil(R)& R\\ 0& 0& nil(R)\end{array}\right)$, and m is a positive integer such that $a_{11}^m=a_{22}^m=a_{33}^m=0$. Then ${\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)}^{2m}=0$. Hence,$$\left(\begin{array}{ccc}nil(R)& 0& 0\\ R& nil(R)& R\\ 0& 0& nil(R)\end{array}\right)\subseteq nil(G_3(R)).$$

Now assume that $\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)\in nil(G_3(R)).$ Then there exists some positive integer m such that ${\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)}^m=0$. Hence $a_{11}^m=a_{22}^m=a_{33}^m=0$, and so$$\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ a_{21}& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right)\in \left(\begin{array}{ccc}nil(R)& 0& 0\\ R& nil(R)& R\\ 0& 0& nil(R)\end{array}\right).$$

Therefore, $nil(G_3(R))=\left(\begin{array}{ccc}nil(R)& 0& 0\\ R& nil(R)& R\\ 0& 0& nil(R)\end{array}\right)$. Then by analogy with the proof of Theorem 2.4, we can show that $G_3(R)$ is nil-M-McCoy.

$(2)\Rightarrow (1)$ it is trivial (see Hashemi, 2013).

In the proof of the next proposition we will need the following lemma.

Lemma 2.4

Zhao et al. (2012, Lemma 2.7) Let R be a semicommutative ring, and $f_1(x),f_2(x),...,f_n(x)$ be in R[x]. If $C_{f_1...f_n}\subseteq nil(R)$, then $C_{f_1}{C}_{f_2}\dots C_{f_n}\subseteq nil(R)$.

Proposition 2.3

Semicommutative rings are nil-M-McCoy rings.

Proof

Let $\alpha ={\sum }_{i=0}^m{a}_i{g}_i$, $\beta ={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0\}$ with $\alpha \beta \in nil(R)[M]$, then we have $C_{\alpha \beta }\in nil(R)$. It follows that $C_{\alpha }{C}_{\beta }\in nil(R)$ by Lemma 2.4. Hence, there exists $r=b_j$ for some $0\le j\le n$ such that $a_{i}r\in nil(R)$ with $0\le i\le m$. This implies that R is a right nil-M-McCoy. Similarly, we can show that R is left nil-M-McCoy. Therefore, R is a nil-M-McCoy.

We now have the following description of the rings which shows one way to give more nil-M-McCoy rings from old ones.

Proposition 2.4

Let $\mathrm{\Omega }$ be an index set and $\{R_I|I\in \mathrm{\Omega }\}$ a family of rings. If $R={\prod }_{I\in \mathrm{\Omega }}{R}_I$, then R is right nil-M-McCoy if and only if every $R_I$ is right nil-M-McCoy for each $I\in \mathrm{\Omega }$.

Proof

It is straightforward to verify that if R is right nil-M-McCoy, then every $R_I$ is right nil-M-McCoy for each I. Conversely, if $\alpha \beta \in nil(R)[M]$ for $\alpha ={\sum }_{i=0}^m{a}_i{g}_i$, $\beta ={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0$}, where $a_i={(a_{i_I})}_{I\in \mathrm{\Omega }}$, $b_j={(b_{j_I})}_{I\in \mathrm{\Omega }}\in R$. For each $I\in \mathrm{\Omega }$, let ${\alpha }_I={\sum }_{i=0}^m{a}_{i_I}{g}_i$, ${\beta }_I={\sum }_{j=0}^n{b}_{j_I}{h}_j\in R_I[M]$, then ${\alpha }_I{\beta }_I\in nil(R_I)[M]$. Since $\beta \ne 0$ there exists some index J with ${\beta }_J\ne 0$. In particular, there exists some nonzero $r_J\in R_J$ with $a_{i_J}{r}_J\in nil(R_J)$ for all $0\le i\le m$ by the nil-M-McCoy property of $R_J$. Fix $r_J\in R_J-\{0\}$, and take r to be the sequence with $r_J$ in the $J^{\mathit{th}}$ coordinate and zero elsewhere. Clearly $a_{i}r\in nil(R)$ and $r\ne 0$.

Corollary 2.2

A finite direct product of right nil-M-McCoy rings is right nil-M-McCoy.

Theorem 2.5

For any monoid M. Then we have the following statements:

(1)	nil-M-Armendariz rings are nil-M-McCoy;
(2)	M-McCoy rings are nil-M-McCoy;
(3)	M-Armendariz rings are nil-M-McCoy.

Proof

(1) Let R be a nil-M-Armendariz rings and $\alpha =a_1{g}_1+\cdots +a_m{g}_m$, $\beta =b_1{h}_1+\cdots +b_n{h}_n\in R[M]-\{0\}$ satisfy $\alpha \beta \in nil(R)[M]$. Then $a_i{b}_j\in nil(R)$ for each i, j. Since $\alpha \ne 0$ and $\beta \ne 0$, for $1\le l\le n$, $1\le k\le m$ there exists $r=b_l$, $s=a_k\in R-\{0\}$. Hence, $a_{i}r\in nil(R)$ and $s{b}_j\in nil(R)$. Therefore, R is nil-M-McCoy.

(2) It follows easily form the definition.

(3) It is easy to see that each M-Armendariz rings is M-McCoy and therefore nil-M-McCoy by (2).

Let $R_i$ be a ring, for each $i\in Z$, and let $R=\prod R_i$ and $⟨{⨁}_{i\in Z}{R}_i,1_R⟩$ be the subring generated by ${⨁}_{i\in Z}{R}_i$ and $\{1_R\}$. Then we have the following result (see Alhevaz & Moussavi, 2010).

Proposition 2.5

Let $R_i$ be a ring, $R={\prod }_{i\in Z}{R}_i$ and $S=⟨{⨁}_{i\in Z}{R}_i,1_R⟩$. Then R is right nil-M-McCoy if and only if each $R_i$ is right nil-M-McCoy if and only if S is right nil-M-McCoy.

Proof

Let each $R_i$ be a right nil-M-McCoy ring and $\alpha ={\sum }_{i=0}^m{a}_i{g}_i$ and $\beta ={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0\}$ such that $\alpha \beta \in nil(R)[M]$, where $a_i=(a_i^k)$ and $b_j=(b_j^k)$. If there exists $t\in \mathcal{Z}$ such that $a_i^t=0$, for each $0\le i\le m$, then we have $a_{i}c\in nil(R)$, where $c=(0,\dots ,0,1_{R_t},0,\dots )$. Now, suppose for each $k\in \mathcal{Z}$, there exists $0\le i_k\le m$ such that $a_{i_k}^k\ne 0$. Since $\beta \ne 0$, there exists $t\in \mathcal{Z}$ and $0\le j_t\le n$ such that $b_{j_t}^t\ne 0$. Consider ${\alpha }_t={\sum }_{i=0}^t{a}_i^t{g}_i$ and ${\beta }_t={\sum }_{j=0}^t{b}_j^t{h}_j\in R_t[M]-\{0\}$. We have $\alpha \beta \in nil(R)[M]$. Thus, there exists some nonzero $c_t\in R_t$ such that $a_i^t{c}_t\in nil(R)$, for each $0\le i\le m$. Therefore, $a_{i}c\in nil(R)$, for each $0\le i\le m$, where $c=(0,\dots ,0,c_t,0,\dots )$, and so R is right nil-M-McCoy.

Conversely, suppose R is right nil-M-McCoy, $t\in Z$, $\alpha ={\sum }_{i=0}^m{a}_i{g}_i$ and $\beta ={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0\}$ such that $\alpha \beta \in nil(R)[M]$. Let ${\alpha }^{\prime }={\sum }_{i=0}^m(1,...,1,a_i,1,...)g_i$ and ${\beta }^{\prime }={\sum }_{j=0}^n(0,...,0,b_j,0,...)h_j\in R[M]-\{0\}$. So there exists $0\ne c=(c_i)$ and $(1,\dots ,1,a_i,1,\dots )c\in nil(R)$ clearly, $c=(0,\dots ,0,c_t,0,\dots )$. Thus, we have $a_i{c}_t\in nil(R_t)$ and so $R_t$ is right nil-M-McCoy. It is easy to see that S is right nil-M-McCoy if and only if each $R_i$ is right nil-M-McCoy.

Let $(M,\le )$ be an ordered monoid. If for any $g_1,g_2,h\in M$, $g_1<g_2$ implies that $g_{1}h<g_{2}h$ and $g_{2}h<g_{1}h$, then $(M,\le )$ is called a strictly totally ordered monoid.

Corollary 2.3

Let M be a strictly totally ordered monoid and R a reversible ring. Then R is nil-M-McCoy.

It was shown in Liu (2005, Proposition 1.4), that if M is strictly totally ordered monoid and I is a reduced ideal of R such that R / I is an M-Armendariz ring, then R is M-Armendariz. The following result generalizes this.

Theorem 2.6

Let M be a strictly totally ordered monoid and I an ideal of R. If I is semicommutative (as a ring without identity) and R / I is nil-M-McCoy, then R is nil-M-McCoy.

Proof

Let $\alpha =a_1{g}_1+\cdots +a_n{g}_n$, $\beta =b_1{h}_1+\cdots +b_m{h}_m\in R[M]-\{0\}$ be such that $\alpha \beta \in nil(R)[M]$ and $g_1<g_2<...<g_n$, $h_1<h_2<...<h_m$. We will use transfinite induction on the strictly totally ordered set $(M,\le )$ to show that $a_{i}r\in nil(R)$ for some nonzero $r\in R$ and for each i. Note that in (R / I)[M], $({\overline{a}}_1{g}_1+{\overline{a}}_2{g}_2+\cdots +{\overline{a}}_n{g}_n)({\overline{b}}_1{h}_1+{\overline{b}}_2{h}_2+\cdots +{\overline{b}}_m{h}_m)\in nil(R/I)[M]$. The fact that R / I is nil-M-McCoy means that there exists r in R such that $\overline{a_i}r\in nil(R/I)$ thus, there exists $p\in N$ such that ${(a_{i}r)}^p\in I$ for each i. If there exists $1\le i\le n$ and $1\le j\le m$ such that $g_i{h}_j=g_1{h}_1$, then $g_1\le g_i$ and $h_1\le h_j$. If $g_1<g_i$, then $g_1{h}_1<g_i{h}_1\le g_i{h}_j=g_1{h}_1$, a contradiction. Thus, $g_i=g_1$. Similarly, $h_j=h_1$. Therefore $a_{i}r\in nil(R)$ for some nonzero $r\in R$ and for each i.

Now suppose that $\omega \in M$ is such that $a_{i}r\in nil(R)$ for each $g_i$ and $h_j$ with $g_i{h}_j=\omega$. Set $X=\{(g_i,h_j)|g_i{h}_j=\omega \}$. Then X is a finite set. We write X as $\{(g_{i_t},h_{j_t})|t=1,\dots ,k\}$ such that $g_{i_1}<g_{i_2}<\dots <g_{i_k}$. Since M is cancellative, $g_{i_1}=g_{i_2}$ and $g_{i_1}{h}_{j_1}=g_{i_2}{h}_{j_2}=\omega$ imply $h_{j_1}=h_{j_2}$. Since $g_{i_1}<g_{i_2}$ and $g_{i_1}{h}_{j_1}=g_{i_2}{h}_{j_2}=w$, we have $h_{j_2}<h_{j_1}$. Thus $h_{j_k}<...<h_{j_2}<h_{j_1}$.

Now ${\sum }_{(g_i,h_j)\in X}{a}_{i}r={\sum }_{t=1}^k{a}_{i_t}r\in nil(R)$. For any $t\ge 2$, $g_{i_1}{h}_{j_t}<g_{i_t}{h}_{j_t}=w$, and thus by induction hypothesis, we have $a_{i_1}r\in nil(R)$. For $t=2$, let ${(a_{i_1}r)}^q=0$. Then ${(r{a}_{i_1})}^{q+1}=0$. Thus$$((a_{i_2}r){(a_{i_1}r)}^{p+1}{a}_{i_2}){(r{a}_{i_1})}^{q+1}(r{(a_{i_1}r)}^{p+1})=0.$$

Since

$((a_{i_2}r){(a_{i_1}r)}^{p+1}{a}_{i_2})r{a}_{i_1})\in I,$

${(r{a}_{i_1})}^q(r{(a_{i_1}r)}^{p+1})\in I,$

$r{(a_{i_1}r)}^p{a}_{i_2}\in I,$

and I is semicommutative, it follows that$$((a_{i_2}r){(a_{i_1}r)}^{p+1}{a}_{i_2})(r{a}_{i_1})(r{(a_{i_1}r)}^p)a_{i_2}){(r{a}_{i_1})}^q(r{(a_{i_1}r)}^{p+1})=0,$$

that procedure yields that ${[(a_{i_2}r){(a_{i_1}r)}^{p+1}]}^{q+3}=0$. Thus $(a_{i_2}r){(a_{i_1}r)}^{p+1}\in nil(I)$.

Since ${[{(a_{i_1}r)}^{p+1}(a_{i_2}r)]}^{q+4}=0$ and $(a_{i_2}r){(a_{i_1}r)}^{p+1}\in nil(I)$, we have ${(a_{i_1}r)}^{p+1}(a_{i_2}r)\in nil(I)$. Similarly one can show that $(a_{i_t}r){(a_{i_1}r)}^{p+1}\in nil(I)$ and ${(a_{i_1}r)}^{p+1}(a_{i_t}r)\in nil(I)$, for$t=2,3,...,k$.

Since $0={[(a_{i_2}r){(a_{i_1}r)}^{p+1}]}^{q+3}=[(a_{i_2}r){(a_{i_1}r)}^{p+1}]...[(a_{i_2}r){(a_{i_1}r)}^{p+1}]$ and ${(a_{i_1}r)}^{p+1}(a_{i_3}r)\in I$ and I is semicommutative, we have$$0=[(a_{i_2}r){(a_{i_1}r)}^{p+1}][{(a_{i_1}r)}^{p+1}(a_{i_3}r)][(a_{i_2}r){(a_{i_1}r)}^{p+1}\times ...\times$$$$[(a_{i_2}r){(a_{i_1}r)}^{p+1}][{(a_{i_1}r)}^{p+1}][{(a_{i_1}r)}^{p+1}(a_{i_3}r)][(a_{i_2}r){(a_{i_1}r)}^{p+1}].(1)$$

Multiplying (1) on the right side by ${(a_{i_1}r)}^{p+1}$ and on the left side by $a_{i_3}r$, then$$\begin{array}{cc}& {[(a_{i_3}r)(a_{i_2}r){(a_{i_1}r)}^{2s+2}]}^{q+3}=0.\text{Hence}\hfill \\ \hfill & (a_{i_3}r)(a_{i_2}r){(a_{i_1}r)}^{2s+2}\in nil(I).\hfill \end{array}$$

Similarly one can show that$$(a_{i_{r_s}}r)(a_{i_{r_{s-1}}}r)...(a_{i_{r_2}}r){(a_{i_{r_1}}r)}^{sp+s}\in nil(I),$$

for each $s\ge 2$ and $\{r_1,...,r_s\}\ne 1$. Since ${\sum }_{(g_i,h_j)\in X}{a}_{i}r={\sum }_{t=1}^k{a}_{i_t}r\in nil(R)$, there exists $s\in N$ such that ${(\sum a_{i_t}r)}^s=0$. Clearly, ${({\sum }_{t=1}^k{a}_{i_t}r)}^s$ equal to sums of such elements $(a_{i_1}r)(a_{i_2}r)...(a_{i_{r_s}}r)$, where $r_t\in \{1,...,k\}$ for each t. Then$${(a_{i_1}r)}^s=-\sum (a_{i_1}r)(a_{i_2}r)...(a_{i_{r_s}}r)(2)$$

where $r_t\in \{1,...,k\}$ and $\{r_1,...,r_s\}\ne 1$. Multiplying Equation (2) on the right side by ${(a_{i_1}r)}^{sp+s}$, we have$${(a_{i_1}r)}^{sp+2s}=-\sum (a_{i_1}r)(a_{i_2}r)...(a_{i_{r_s}}r){(a_{i_1}r)}^{sp+s}\in nil(I)$$

where $r_t\in \{1,...,k\}$ and $\{r_1,...,r_s\}\ne 1$, since nil(I) is an ideal of I by Lunqun and Jinwang (2013, Lemma 3.1). Hence $(a_{i_1}r)\in nil(I)$.

Since ${(a_{i_2}r)}^p\in I$, by analogy with the above proof we have that $(a_{i_t}r){(a_{i_2}r)}^{p+1}\in nil(I)$ for each $3\le t\le k$. Since $(a_{i_1}r)$ and $({\sum }_{t=1}^k{a}_{i_t}r)$ are nilpotent, there exists $s\in N$ such that ${(a_{i_1}r)}^s=0={({\sum }_{t=1}^k{a}_{i_t}r)}^s$. Then ${(a_{i_2}r)}^{p+1}{(a_{i_1}r)}^s{(a_{i_2}r)}^{p+1}=0$. Since ${(a_{i_2}r)}^{p+1}\in I$ and I is semicommutative, we have ${((a_{i_1}r){(a_{i_2}r)}^{p+1})}^{s+1}=0$. Thus $(a_{i_1}r)(a_{i_1}r)\in nil(I)$. Hence $(a_{i_t}r){(a_{i_2}r)}^{p+1}\in nil(I)$ for each $t\ne 2$. By analogy with the above proof we have $(a_{i_{r_s}}r)(a_{i_{r_{s-1}}}r)...(a_{i_{r_2}}r){(a_{i_{r_1}}r)}^{sp+s}\in nil(I)$, where $s\ge 2$ and $\{r_1,...,r_s\}\ne 2$. Since ${(a_{i_1}r)}^s=0={({\sum }_{t=1}^k{a}_{i_t}r)}^s$, we have$${(a_{i_2}r)}^s=-\sum (a_{i_{r_1}}r)(a_{i_{r_2}}r)...(a_{i_{r_s}}r)(3)$$

where $r_t\in \{1,...,k\}$, $\{r_1,...,r_s\}\ne 1$ and $\{r_1,...,r_s\}\ne 2$. Multiplying Equation (3) on the right side by ${(a_{i_2}r)}^{sp+s}$, then ${(a_{i_2}r)}^{sp+s}=-\sum (a_{i_{r_1}}r)(a_{i_{r_2}}r)...(a_{i_{r_s}}r){(a_{i_2}r)}^{sp+s}\in nil(I)$ where $r_t\in \{1,...,k\}$, $\{r_1,...,r_s\}\ne 1$ and $\{r_1,...,r_s\}\ne 2$. Therefore $a_{i_2}r\in nil(I)$. Similarly one can show that $a_{i_3}r,...,a_{i_k}r$ are nilpotent. Then $a_{i}r\in nil(I)$ for some nonzero $r\in R$ and for each i. This implies that R is a right nil-M-McCoy. Similarly, we can show that R is left nil-M-McCoy. Therefore, R is a nil-M-McCoy.

The following example shows that the condition R / I is a nil-M-McCoy ring in Theorem 2.6 is not superfluous.

Example 2.5

Let $M=(N\cup \{0\},+)$, and let F be any field and $R=Z_2\oplus M_2(F)$. Then $I=Z_2\oplus 0$ is a semicommutative ideal of R, but $R/I\cong M_2(F)$ is not nil-M-McCoy ring. Let $\alpha =(1,E_{11})+(0,E_{12})g_1+(1,E_{21})g_2+(1,E_{22})g_3$ and $\beta =(0,-E_{21})+(0,E_{11})h_1+(0,E_{22})h_2+(0,-E_{12})h_3\in R[M]-\{0\}$. Then $\alpha \beta =0$, but if $r\in R$ and $(1,E_{\mathit{ij}})r\in nil(R)$, for each i,j then clearly $r=0$. Therefore, R is not right nil-M-McCoy, and hence it is not nil-M-McCoy.

The following lemma shows that the condition M is an "u.p."-monoid in Theorem 2.7 is not superfluous.

Lemma 2.5

Let M be a cyclic group of order $n\ge 2$ and R any NI ring. Then R is not nil-M-McCoy.

Proof

Suppose that $M=\{e,h,h^2,...,h^{n-1}\}$. Let $\alpha =1e+1h+1h^2+\cdots +1h^{n-1}$ and $\beta =1e+(-1)h$. Then $\alpha \beta =0$. Therefore, R is not nil-M-McCoy, whereas R is NI ring.

A ring R is a subdirect sum of a family of rings ${\{R_i\}}_{i\in I}$ if there is a surjective homomorphism, where ${\pi }_k:{\prod }_{i\in I}{R}_i\to R_k$ is the $k^{\mathit{th}}$ projection.

Now we consider the case of subdirect sum of nil-M-McCoy.

Proposition 2.6

If R is a subdirect sum of nil-M-McCoy then R is nil-M-McCoy.

Proof

Let $I_k$ for each $k\in \{1,...,l\}$ be ideals of R and for them ring $R/I_k$, is nil-M-McCoy and let ${\bigcap }_{k=1}^l{I}_k=0$. Let $\alpha ={\sum }_{i=0}^m{a}_i{g}_i$ and $\beta ={\sum }_{j=0}^n{b}_j{h}_j\in R[M]-\{0\}$ such that $\alpha \beta \in nil(R)[M]$. Since $R/I_k$, for each k, is nil-M-McCoy then we have ${(a_{i}r)}^{r_i{k}_j}\in I_k$$\forall i,j$. Now let $r_{\mathit{ij}}=\{r_i{k}_j|k=1,...,l\}$, therefore ${(a_{i}r)}^{r_{\mathit{ij}}}\in {\bigcap }_{k=1}^l{I}_k=0$. Since $a_{i}r\in nil(R)$, for each i. Hence, R is nil-M-McCoy.

The ring of Laurent polynomials in x with coefficients in a ring R, consists of all formal sums ${\sum }_{i=k}^n{r}_i{x}^i$ with obvious addition and multiplication, where $r_i\in R$ and k, n are (possibly negative) integers, denoted by $R[x,x^{-1}]$. We finish this paper with the following corollary.

Corollary 2.4

Let R be a semicommutative ring. Then R is nil-$Z$-McCoy, for any $\alpha =a_{-m}{x}^{-m}+a_{-(m-1)}{x}^{-(m-1)}+\cdots +a_p{x}^p$, $\beta =a_{-n}{x}^{-n}+a_{-(n-1)}{x}^{-(n-1)}+\cdots +a_q{x}^q\in R[x,x^{-1}]-\{0\}$, if $\alpha \beta \in nil(R)[x,x^{-1}]$, then there exist $r,s\in R-\{0\}$ such that $a_{i}r\in nil(R)$ and $s{b}_j\in nil(R)$ for each $-m\le i\le p$ and $-n\le j\le q$.

Proof

Note that $R[Z]\cong R[x,x^{-1}]$ (see Alhevaz, Moussavi, & Habibi, 2012).

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Vahid Aghapouramin

Vahid Aghapouramin is currently a PhD student at the department of mathematics, Karaj Branch, Islamic Azad university, Karaj, 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, Non-commutative Algebra, Algebraic Combinatorics.