Generalized f-semiperfect Modules

Abstract

In this paper we introduce generalized f-semiperfect modules as a generalization of the generalized semiperfect modules. We give various properties of the generalized f-semiperfect modules. We show that: (i) every generalized f-semiperfect module over a regular ring is f-semiperfect; (ii) for small or for finitely generated submodules $L$ of $M$, the factor module $\frac{M}{L}$ is generalized f-semiperfect; (iii) If $M$ is a projective and generalized f-semiperfect module such that every $Rad$-supplement in $M$ is a direct summand of $M$, then every direct summand of $M$ is a generalized f-semiperfect module; (iv) If $M=\oplus_{i\in I}M_{i}$ is a locally Noetherian and duo module such that $\{M_{i}\}_{i\in I}$ is the family of generalized f-semiperfect modules, then $M$ is a generalized f-semiperfect module.