No abstract

ABSTRACT. The generation of proper classes of short exact sequences of modules by subclasses s considered. The class generated by two proper classes is studied by means of some operations between these classes These operauons are investigated m details for classes of short quasi-splitting, torsionsphtting and pure exact sequences of abelian groups.

No abstract

We study the properties of˚-cofinitely radical supplemented modules, or, briefly, cgs˚-modules. It is shown that a module with summand sum property (SSP) is cgs˚if and only if M=w Loc˚M .w Loc˚M is the sum of all w-local direct summands of a module M / does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs˚-module is cgs˚; and that, for any ring R; every free R-module is cgs˚if and only if R is semiperfect.

It is well known that a projective module M is ⊕-supplemented if and only if M is semiperfect. We show that a projective module M is ⊕-cofinitely supplemented if and only if M is cofinitely semiperfect or briefly cof-semiperfect (i.e., each finitely generated factor module of M has a projective cover). In this paper we give various properties of the cof-semiperfect modules. If a projective module M is semiperfect then every M -generated module is cof-semiperfect. A ring R is semiperfect if and only if every free R-module is cof-semiperfect.