A COUNTEREXAMPLE FORCS-RINGS

Abstract

A module M iscalled a CS-module or an extending module if every submodule is essentialin a direct summand of M. A ring R is called a rightCS-ring or a right extending ring if R_R is a CS-module.For several types of right CS-rings it is known that either all right ideals or some large class ofright ideals inherit the CS property. For example, by a result of Dung-Smith or Vanaja-Purav, a ringR is (right and left) Artinian, serial, and J(R)^2 = 0 if andonly if every R-module is CS. In particular, if R is a QF-ringand J(R)^2 = 0 (hence R is serial), then everyR-module is CS. However we exhibit a finite, serial, strongly bounded QF groupalgebra R with J(R)^3 = 0 for which there is aprincipal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.