Stability of rings

TL;DR

This paper investigates the stability conditions of elements in linear groups over various associative rings, linking ring stability to classical algebraic results and examining specific classes of rings.

Contribution

It provides a unified analysis of ring stability conditions, connecting classical results with new insights on specific ring classes.

Findings

Stability conditions for commutative rings analyzed

Connection between ring stability and linear group elements established

Classical results interpreted from a unified perspective

Abstract

The conditions for stability of the elements of linear groups over the associative rings with identity and their connection with the stability of rings are analyzed in the article. The stability of rings which are commutative, satisfy the conditions of stability of rank $\geq 2$, von Neumann regular, integer-algebraic, nearly local rings introduced by the author, is examined. The most important classical results of H. Bass, L. Vaserstein, S.H. Khlebutin, A.A. Suslin, J.S. Wilson, I.Z.Golubchik, are considered from the unified standpoint.