Surjections of unit groups and semi-inverses

TL;DR

This paper investigates conditions under which surjective ring homomorphisms induce surjective maps on unit groups, introducing generalized inverses and specific ring classes to characterize these cases.

Contribution

It introduces notions of generalized inverses and units, and identifies classes of rings where surjections induce surjections on unit groups.

Findings

Surjections from certain rings induce surjective maps on unit groups.

Introduction of generalized inverses and units concepts.

Characterization of rings with closed sets of spectrum points.

Abstract

Given a surjective ring homomorphism, we study when the induced group homomorphism on unit groups is surjective. To this end, we introduce notions of generalized inverses and units, as well as a class of rings such that the set of closed points in the spectrum is a closed set. It is shown that any surjection out of such a ring induces a surjection on unit groups.