Extension of Almost Armendariz Rings

Abstract

A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{\ast }(R)$. In this article, for an endomorphism $\alpha$ on $R$, we define an $\alpha$-almost Armendariz ring of $R$ considering the polynomials in skew polynomial ring $R[x;\alpha ]$ instead of $R[x]$. It is the generalisation of an almost Armendariz ring [9] and an $\alpha$-Armendariz ring [4]. Moreover, for an endomorphism $\alpha$ of $R$, we define an $\alpha$-skew almost Armendariz ring, and prove that a reversible ring $R$ with certain condition on endomorphism $\alpha$, its polynomial ring $R[x]$ is an $\overline{\alpha }$-skew almost Armendariz ring.