Subrings which are closed with respect to taking the inverse

TL;DR

This paper explores conditions under which subrings maintain the unit group structure of the larger ring, with a focus on matrix rings and subrings defined by reflexive and transitive relations.

Contribution

It provides new insights into when the intersection of a subring with the units of the larger ring equals the units of the subring, especially in matrix rings with structural subrings.

Findings

In many cases, S ∩ U(R) = U(S) holds.

Special emphasis on matrix rings and reflexive, transitive relation-defined subrings.

Results clarify when subrings preserve unit group structures.

Abstract

Let S be a subring of the ring R. We investigate the question of whether S intersected by U(R) is equal to U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.