Bezout Rings, Polynomials, and Distributivity

Abstract

Let A be a ring, ϕ be an injective endomorphism of A, and let $A_r \left[ {x,\varphi } \right] \equiv R$ be the right skew polynomial ring. If all right annihilator ideals of A are ideals, then R is a right Bezout ring $\Leftrightarrow A$ is a right Rickartian right Bezout ring, ϕ(e)=e for every central idempotent e∈A, and the element ϕ(a) is invertible in A for every regular a∈A. If A is strongly regular and n≥ 2, then R/x ⁿ R is a right Bezout ring $\Leftrightarrow$ R/x ⁿ R is a right distributive ring $\Leftrightarrow$ R/x ⁿ R is a right invariant ring $\Leftrightarrow$ ϕ(e)=e for every central idempotent e∈A. The ring R/x ² R is right distributive $\Leftrightarrow$ R/x ⁿ R is right distributive for every positive integer n $\Leftrightarrow$ A is right or left Rickartian and right distributive, ϕ(e)=e for every central idempotent e∈A and the ϕ(a) is invertible in A for every regular a∈A. If A is a ring which is a finitely generated module over its center, then A[x] is a right Bezout ring $\Leftrightarrow$ A[x]/x ² A[x] is a right Bezout ring $\Leftrightarrow$ A is a regular ring.