Bezout Rings, Polynomials, and Distributivity

Let A be a ring, ϕ be an injective endomorphism of A, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A_r \left[ {x,\varphi } \right] \equiv R$$ \end{document} be the right skew polynomial ring. If all right annihilator ideals of A are ideals, then R is a right Bezout ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow A$$ \end{document} 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/xnR is a right Bezout ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}R/xnR is a right distributive ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}R/xnR is a right invariant ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document} ϕ(e)=e for every central idempotent e∈A. The ring R/x2R is right distributive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}R/xnR is right distributive for every positive integer n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}A[x]/x2A[x] is a right Bezout ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \Leftrightarrow $$ \end{document}A is a regular ring.