AI 中文总结
本文受Cline公式启发，用群可逆和EP元素刻画可逆环与$\ast$-可逆环，证明两个等价条件，并构造反例区分环类及讨论包含关系。
AI 中文摘要
本文研究环中两个元素的反向乘积性质。受Cline公式的启发，我们利用群可逆元素、EP元素以及广义逆在反向乘积下的传递行为来刻画可逆环和$\ast$-可逆环。我们证明单位环$R$是可逆环当且仅当$ab\in R^{\sharp}$蕴含$ba\in R^{\sharp}$。对于对合环$R$，$R$是$\ast$-可逆环恰好当$ab\in R^{\mathrm{EP}}$蕴含$b^{\ast}a\in R^{\mathrm{EP}}$。我们构造了若干反例以区分这些环类，并讨论了它们之间的相互包含关系。
英文摘要
This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and $\ast$-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring $R$ is reversible if and only if $ab\in R^{\sharp}$ yields $ba\in R^{\sharp}$. For an involutive ring $R$, $R$ is $\ast$-reversible precisely whenever $ab\in R^{\mathrm{EP}}$ implies $b^{\ast}a\in R^{\mathrm{EP}}$. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.
