Strongly exchange rings

TL;DR

This paper introduces strongly right exchange rings, a new class characterized by descending chains of right coprime pairs, unifying various ring classes and revealing their semiregularity and structural properties.

Contribution

It defines strongly right exchange rings via descending chains of right coprime pairs and explores their properties, unifying several known classes of rings.

Findings

Strongly right exchange rings are semiregular.

They include left injective, pure-injective, cotorsion, local, and left continuous rings.

Descending chains of right coprime pairs characterize these rings.

Abstract

Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of strongly right exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprime pairs.