Notes on Reversed Products of Two Elements in Rings

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 $ab\in R^{\mathrm{♯}}$ yields $ba\in R^{\mathrm{♯}}$. For an involutive ring $R$, $R$ is $\ast$-reversible precisely whenever $ab\in R^{\mathrm{E}\mathrm{P}}$ implies $b^{\ast }a\in R^{\mathrm{E}\mathrm{P}}$. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.