Quasi-Frobenius rings and Nakayama permutations of semiperfect rings

We say that A is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) A - module U, the dual module U* is a simple left (right) A-module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles. © 2002 Plenum Publishing Corporation.