Extending Modules Satisfying (S*)
by
Ayse Cigdem Ozcan
Hacettepe University Department of Mathematics 06532 Beytepe Ankara Turkey

Leonard [L] called a (right) R-module M a small module if M a small submodule of its injective hull (i.e. M << E(M)). Then

Z*(M)={ m in M : mR     is a small module }

defined by Harada [H]. We call a module M is a cosingular module if Z^*(M)=M. On the other hand, let K be a class of modules. The class d^*K is defined by Al-Khazzi and Smith [AS] as the class of modules M such that for every submodule N of M there exists a direct summand Kof M such that K <= N and N/K in K. Now we say that a module M satisfies (S^*) if M is in the class d^*K when K is the class of cosingular modules. In this paper we prove that if M is an extending module satisfying (S^*) and Z^*(M) is semisimple projective then every submodule of M is extending and M is a locally Artinian serial SI-module. We also characterize some rings by using (S^*) property. For example,

Theorem 1 The following are equivalent for a ring R.
i) Every injective R-module is lifting (a (right) H-ring).
ii) R is right perfect and every (extending) R-module satisfies (S^*).
iii) R is right perfect and the injective hull of every semisimple module satisfies (S^*).

Theorem 2 The following are equivalent for a ring R.
i) R is right perfect and every R-module with (S^*) is extending.
ii) R is semiperfect with Rad(R^(N)) << R^(N) and every R-module with (S^*) is extending.
iii) R is a generalized uniserial ring with J(R)²=0.

References
[AS] I. Al-Khazzi, P.F. Smith, Classes of modules with many direct summands, J.Austral.Math.Soc., 59, 8-19 (1995).
[H] M. Harada, Non-small modules and non-cosmall modules, In Ring Theory: Proceedings of the 1978 Antwerp Conference, ed.NewYork: Marcel Dekker.
[L] W.W. Leonard, Small modules, Proc.Amer.Math.Soc., 17, 527-531, 1966.

Date received: July 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-43.