Transposed Triple Products and Pro-Symmetric Rings in $\ast$-Rings

Abstract: 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 $abc\in R<sup>{\sharp}$ implies $acb\in R<sup>{\sharp}$ 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.