Mininjective Rings

Abstract

A ringRis called right mininjective if every isomorphism between simple right ideals is given by left multiplication by an element ofR. These rings are shown to be Morita invariant. IfRis commutative it is shown thatRis mininjective if and only if it has a squarefree socle, and that every image ofRis mininjective if and only ifRhas a distributive lattice of ideals. IfRis a semiperfect, right mininjective ring in whicheRhas nonzero right socle for each primitive idempotente, it is shown thatRadmits a Nakayama permutation of its basic idempotents, and that its two socles are equal if every simple left ideal is an annihilator. This extends well known results on pseudo- and quasi-Frobenius rings.