In this paper, we introduce the concept of (amply) cofinitely ss-supplemented modules as a proper generalization modules, and provide various properties these modules. particular, prove that arbitrary sum is ss-supplemented. Moreover, show ring R semiperfect Rad(R)⊆Soc(RR) if only every left R-module

‎We call a ring $R$ right generalized semiperfect if every simple right $R$-module is an epimorphic image of a flat right $R$-module with small kernel‎, ‎that is‎, ‎every simple right $R$-module has a flat $B$-cover‎. ‎We give some properties of such rings along with examples‎. ‎We introduce flat strong covers as flat covers which are also flat $B$-covers and give characterizations of $A$-perfe...

Let M be a left module over a ring R and I an ideal of R. We call (P,f ) a projective I -cover of M if f is an epimorphism from P to M , P is projective, Kerf ⊆ IP , and whenever P = Kerf + X, then there exists a summand Y of P in Kerf such that P = Y +X. This definition generalizes projective covers and projective δ-covers. Similar to semiregular and semiperfect rings, we characterize I -semir...

The aim of this paper is to investigate strong notion strongly ⨁-supplemented modules in module theory, namely ⨁-locally artinian supplemented modules. We call a M if it locally and its supplement submodules are direct summand. In study, we provide the basic properties particular, show that every summand supplemented. Moreover, prove ring R semiperfect with radical only projective R-module

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. K. Nicholson [7]. In 2002, V. P. Camillo and D. D. Anderson [1] investigated commutative clean rings and obtained several important results. In [4] Han and Nicholson show that if A is a semiperfect ring, then A[Z2] i...

‎we call a ring $r$ right generalized semiperfect if every simple right $r$-module is an epimorphic image of a flat right $r$-module with small kernel‎, ‎that is‎, ‎every simple right $r$-module has a flat $b$-cover‎. ‎we give some properties of such rings along with examples‎. ‎we introduce flat strong covers as flat covers which are also flat $b$-covers and give characterizations of $a$-perfe...

We study the class of ADS rings and modules introduced by Fuchs [F]. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that RR is ADS must be either right self-injective or indecomposable as a right Rmodule. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of co...

We characterize semiperfect modules, semiperfect rings, and perfect rings using locally projective covers and generalized locally projective covers, where locally projective modules were introduced by Zimmermann-Huisgen and generalized locally projective coves are adapted from Azumaya’s generalized projective covers.