Geodesic
A COUNTEREXAMPLE FORCS-RINGS
Glasgow mathematical journal, Tome 42 (2000) no. 2, pp. 263-269

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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. 
BIRKENMEIER, GaryF.; KIM, JIN YONG; PARK, JAE KEOL. A COUNTEREXAMPLE FORCS-RINGS. Glasgow mathematical journal, Tome 42 (2000) no. 2, pp. 263-269. doi: 10.1017/S0017089500020127
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