Automorphism-invariant non-singular rings and modules

Abstract

$\text{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\text{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\text{2)}$ $A$ is a right automorphism-invariant regular ring. $\text{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\text{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\text{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\text{2)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an automorphism-invariant module. $\text{3)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an injective module.

9 pages. The study is supported by Russian Scientific Foundation (project 16-11-10013). arXiv admin note: text overlap with arXiv:1701.07117