A COUNTEREXAMPLE FORCS-RINGS

Abstract

A module M is called a CS-module or an extending module if every submodule is essential in a direct summand of M. A ring R is called a right CS-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 of right 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 and only if every R-module is CS. In particular, if R is a QF-ring and J(R)^2 = 0 (hence R is serial), then everyR -module is CS. However we exhibit a finite, serial, strongly bounded QF group algebra R with J(R)^3 = 0 for which there is a principal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.