Symmetry of extending properties in nonsingular Utumi rings

TL;DR

This paper explores the symmetrical properties of nonsingular Utumi rings, establishing equivalences between right and left conditions and extending these properties to modules, with implications for ring classification.

Contribution

It demonstrates the right-left symmetry of CS and max-min CS conditions in nonsingular rings and extends these results to nonsingular modules, providing new characterizations.

Findings

Right-left symmetry of CS and max-min CS conditions in nonsingular rings

Equivalence of ring properties under right and left conditions

Characterization of semiprime nonsingular rings with finite uniform dimension

Abstract

This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp. right min, right max-min) CS if and only if it is left min (resp. left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.