Some remarks on surjections of unit groups

Abstract

. The main purpose of this paper is to study unital ring homomorphisms of associative rings ϕ : R → S satisfying one of the following conditions: (a) the unit-preserving property: ϕ ( R × ) = S × and (b) the inverse unit-preserving property: ϕ − 1 ( S × ) = R × . We establish the relationship between these two conditions. Several characterizations of such conditions are settled. An application to the index of unit groups of rings R ⊂ S having a nonzero common ideal is given.