On quasi-Armendariz skew monoid rings

Abstract

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called quasi-Armendariz if whenever $f(x)=Σa_ix^i$ and $g(x)=Σb_jx^j$ in $R[x]$ satisfy $f(x)R[x]g(x)=0$, we have $a_iRb_j=0$ for every $0leq i leq m$ and $0leq j leq n$) and provide rich classes of non-semiprime quasi-Armendariz rings. <br /><br />Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called quasi-Armendariz if whenever $f(x)=Σa_ix^i$ and $g(x)=Σb_jx^j$ in $R[x]$ satisfy $f(x)R[x]g(x)=0$, we have $a_iRb_j=0$ for every $0leq i leq m$ and $0leq j leq n$) and provide rich classes of non-semiprime quasi-Armendariz rings.