Strongly rational comodules and semiperfect Hopf algebras over QF rings

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. U sing this notion a number of characterizations of right semiperfect coalgeb ras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show tha t a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or - equivalently - the (left) integrals form a free R- module of rank 1. (C) 2001 Elsevier Science B.V. All rights reserved. MSG: 16W30; 16L60; 16D90.