In this paper, we describe $ss$-supplement submodules in terms of a special class endomorphisms. Let $R$ be ring with semisimple radical and $P$ projective $R-$module. We show that there is bijection between ss-supplement $End_{R}(P)$. Moreover, define radical-s-projective modules as generalization modules. prove every submodule $R-$module over the radical. $SSI$-ring $R$, projective. provide r...

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this ...

We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules.

Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-...

A right module M over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper we find a suitable condition under which a special ω-elongation of a summable QTAG-module by a ( ω +k)-projective QTAG-module is also a summable QTAG-module.

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Ext-orthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules.