ON RINGS SATISFYING BOTH OF 1-abc AND 1-cba BEING INVERTIBLE OR NONE

Abstract

Let $R$ be a ring with identity 1 and $n$ a positive integer. We define the property Pn as  follows:

(Pn) If $1- a_1a_2a_3 \cdots a_{n-1}a_n$ is invertible in $R$, then so is $1- a_na_2a_3 \cdots a_{n-1}a_1$.

Thus, $R$ satisfies (Pn), for some $n \ge 3$ if and only if $R$ satisfies (P3). Some properties of rings satisfying (P3) are obtained, e.g., $R$ must be directly finite.