Unit lifting morphisms

Abstract

In this paper, we introduce the concept of a unit lifting morphism, a natural generalization of the classical notion “local morphism”, by providing several examples and investigating all its properties and interrelations with local morphisms and unit lifting ideals. Moreover, we expose a relation between a unit lifting morphism f:R→S of rings and the pair (ker(f),f−1(U(S))) and show that unit lifting morphisms correspond to the least element of a partially ordered set. As a prominent result, we provide the equivalent conditions on the existence of a unit lifting morphism from a ring R into a semisimple artinian ring which is intimately related to a deep result by Camps and Dicks that characterizes semilocal rings in terms of local morphisms. This result gives rise to the study of a new class of rings, which we call weakly semilocal rings.