Weak e–symmetric rings

Abstract

We show that R is a weak $e-$ symmetric ring if and only if $eR(1-e)Re=0$ and eRe is a symmetric ring. Further, we obtain R is weak $e-$ symmetric if and only if for any $a_1,a_2,\dots ,a_n\in R$, $a_1{a}_2\cdots a_n=0$ implies $e{a}_{\sigma (1)}{a}_{\sigma (2)}\cdots a_{\sigma (n)}e=0$, where $\sigma$ is any transformation of $\{1,2,\dots n\}$. As an application, we show that a ring R is a left min-abel ring if and only if R is a weak $e-$ symmetric ring for any $e\in M{E}_l(R)$. Finally, we show that the definition of $e-$ symmetric ring is not left-right symmetric. Also, with the help of weak $e-$ symmetric rings, we give some characterizations of EP elements.

Keywords:

• $e-$ symmetric ring
• EP element
• idempotent element;
• left min-abel ring
• weak $e-$ symmetric ring
• weak $-EP$ element

2020 Mathematics Subject Classification:

• 16U80
• 16U10
• 15A09
• 16W10
• 16B99

Additional information

Funding

Project supported by the Foundation of Natural Science of China (11471282), Natural Science Fund for Colleges and Universities in Jiangsu Province (11KJB110019; 15KJB110023) and School Foundation of Guangling College (ZKYB180021).