Bezout Rings, Polynomials, and Distributivity

A. A. Tuganbaev

Matematičeskie zametki, Tome 70 (2001) no. 2, pp. 270-288

Résumé

Let $A$ be a ring, $\varphi$ be an injective endomorphism of $A$, and let $A_r[x,\varphi]\equiv R$ be the right skew polynomial ring. If all right annihilator ideals of $A$ are ideals, then $R$ is a right Bezout ring $\iff$ $A$ is a right Rickartian right Bezout ring, $\varphi(e)=e$ for every central idempotent $e\in A$, and the element $\varphi(a)$ is invertible in $A$ for every regular $a\in A$. If $A$ is strongly regular and $n\ge2$, then $R/x^nR$ is a right Bezout ring $R/x^nR$ is a right distributive ring $\iff$ $R/x^nR$ is a right invariant ring $\iff$ $\varphi(e)=e$ for every central idempotent $e\in A$. The ring $R/x^2R$ is right distributive $\iff$ $R/x^nR$ is right distributive for every positive integer $n$ $\iff$ $A$ is right or left Rickartian and right distributive,$\varphi(e)=e$ for every central idempotent $e\in A$ and the $\varphi(a)$ is invertible in $A$ for every regular $a\in A$. If $A$ is a ring which is a finitely generated module over its center, then $A[x]$ is a right Bezout ring $\iff$ $A[x]/x^2A[x]$ is a right Bezout ring $\iff$ $A$ is a regular ring.