Rings with annihilator condition and their extensions

Source abstract

Let $R$ be an associative unital ring and $\alpha$ an endomorphism of $R$. In this article, we aim to investigate rings that satisfy Property $(a.c.)$ and explore some related rings. Moreover, we examine many polynomial extensions of $R$ satisfying Property $(a.c.)$. In particular, $\alpha$-skew quasi-Armendariz rings and $\alpha$-(sps) quasi-Armendariz rings. We prove that this class of rings has always Property $(a.c.)$ on the left but not necessarily on the right. Furthermore, we investigate the transmission of Property $(a.c)$ from a ring $R$ to $R[x,\alpha]$ and $R[[x,\alpha]]$. Received: 1 September 2022 | Accepted: 4 July 2023