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...

Projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimension ≤ 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings R with codimension ≤ 2. Explicit resolutions are...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

We are mainly concerned with the completablity of unimodular rows to an invertible matrix. We first relate this to stable range of the ring. Further, we consider rows generating an idempotent of the ring to obtain completability under special conditions. We try to obtain analogues of important theorems regarding unimodular rows in the case of these “idempotent rows”. By ring, we shall always me...