发表机构
School of Mathematics and Physics, Bengbu University; School of Mathematical Sciences, Yangzhou University; College of Basic Science, Zhejiang Shuren University（蚌埠学院数学与物理学院; 扬州大学数学科学学院; 浙江树人大学基础科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用群可逆元素和EP元素刻画对称环，引入pro-对称环概念，证明其蕴含对称性，并构造反例区分环类，回答开放问题。
AI 中文摘要
本文研究了环中涉及转置三重积的性质。受Cline公式的启发，我们利用群可逆元素和EP元素刻画了对称环。我们证明了一个含单位元的环$R$是对称的当且仅当对于所有$a,b,c\in R$，$abc\in R^{\sharp}$蕴含$acb\in R^{\sharp}$。特别地，我们回答了\cite[问题2.9]{MW1}中提出的问题。我们提供了一个例子来说明对于对称环$R$，$abc=e$不一定推出$acb=e$。对于$\ast$-环，我们引入了pro-对称环的概念：一个环$R$是pro-对称的，如果对于所有$a,b,c\in R$，$abc\in P(R)$蕴含$acb\in P(R)$。我们证明了每个pro-对称环都是对称的。我们构造了几个反例来区分这些环类，并讨论了它们之间的相互包含关系。
英文摘要
This paper investigates properties concerning transposed triple products in rings. Motivated by Cline's formula, we characterize symmetric rings by means of group invertible elements and EP elements. We prove that a unital ring $R$ is symmetric if and only if $abc\in R^{\sharp}$ implies $acb\in R^{\sharp}$ for all $a,b,c\in R$. In particular, we give an answer to the problem posed in \cite[Problem 2.9]{MW1}. An example is provided to illustrate that for a symmetric ring $R$, $abc=e$ does not generally yield $acb=e$. For $\ast$-rings, we introduce the notion of pro-symmetric rings: a ring $R$ is pro-symmetric if $abc\in P(R)$ implies $acb\in P(R)$ for all $a,b,c\in R$. We show that every pro-symmetric ring is symmetric. Several counterexamples are constructed to distinguish these classes of rings, and their mutual inclusion relations are also discussed.
