On Rings with the Goodearl–Menal Condition

Abstract

A ring R is said to satisfy the Goodearl–Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x − u and y − u −1 are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl–Menal condition if and only if no homomorphic image of R is isomorphic to either ℤ2 or ℤ3 or 𝕄2(ℤ2). These results correct two existing results. Some consequences are discussed.