A group G is called morphic if every endomorphism α:G→G for which Gα◃G satisfies G/Gα≅ker (α). Call an endomorphism α∈end(G) regular if αβα=α for some β∈end(G), and call α unit regular if β can be chosen to be an automorphism of G. The main purpose of this paper is to prove the following group-theoretic analogue of a theorem of Ehrlich: if G is a morphic group, an endomorphism α:G→G for which Gα◃G is unit regular if and only if it is regular. As an application, a cancellation theorem is proved that characterizes the morphic groups among those with regular endomorphism monoids.

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

A ring R is called a right SF-ring if all of its simple right R-modules are flat. It is well known that a von Neumann regular ring is a right SF-ring. In this paper we study conditions under which the converse holds.

The purpose of the note is to calculate the group K0(A) where A is a ring all of whose Pierce stalks are simple artinian. This generalizes known results for A self injective of type In. For A regular, with injective hull, Â, of type In, a characterization is given for when A → Â it induces an isomorphism K0(A) → K0(Â).