A generalization of modules with the property (P*)

Burcu Nişancı Türkmen

I.A- Khazzi and P.F. Smith called a module M have the property (P*) if every submodule N of M there exists a direct summand K of M such that $K\leq N$ and $\frac{N}{K}\subseteq Rad(\frac{M}{K})$. Motivated by this, it is natural to introduce another notion that we called modules that have the properties (GP*) and (N - GP*) as proper generalizations of modules that have the property (P*). In this paper we obtain various properties of modules that have properties (GP*) and (N - GP*). We show that the class of modules for which every direct summand is a fully invariant submodule that have the property (GP*) is closed under finite direct sums. We completely determine the structure of these modules over generalized f-semiperfect rings.