AI 中文总结
本文推广Armendariz环，引入右e-Armendariz环概念，证明其无左右对称性并给出例子，核心结论是环为右e-Armendariz环当且仅当e为左半中心元且角环eRe是Armendariz环。
AI 中文摘要
本文引入了e-Armendariz环的概念，这是通过幂等元e对Armendariz环的一种推广。若对于环R中的任意多项式f(x)、g(x)∈R[x]，当满足f(x)g(x)=0时，对f(x)的所有系数a_i和g(x)的所有系数b_j都有a_i b_j e=0，则称环R为右e-Armendariz环。我们证明这类新的环不具有左右对称性，并给出了若干说明性例子。本文的主要结果刻画了：环R为右e-Armendariz环当且仅当幂等元e是左半中心元，且角环eRe是Armendariz环。
英文摘要
This paper introduces the concept of e-Armendariz rings, a generalization of Armendariz rings via idempotent element $e$. A ring $R$ is called right e-Armendariz if for any polynomials $f(x), g(x) \in R[x]$, the condition $f(x)g(x)=0$ implies $a_ib_je=0$ for all coefficients $a_i$ of $f(x)$ and $b_j$ of $g(x)$. We establish that this new class of rings is not left-right symmetric and provide several illustrative examples. The main result of this paper provides a characterization that a ring $R$ is right e-Armendariz if and only if the idempotent $e$ is left semicentral and the corner ring $eRe$ is an Armendariz ring.
