Supplement submodules and a generalization of projective modules

Abstract

Let R be a ring with Jacobson radical J (R). Given a left R-module M, a supplement submodule of M is a submodule K M for which there exists K ≤ M such M = K + K and K is minimal with respect to this property. In general, supplement submodules are strict generalizations of direct summands. Those rings for which every supplement submodule of a finitely generated projective module is a direct summand have been widely treated in the literature (see [6][7][8]11,13,16]). However, it is an open problem to give an internal characterization of these rings (see [9]). The purpose of this note is to give more results about supplement submodules in projective modules and determine their relationship with a generalization of projective modules (radical-projective modules). These results will allow us to generalize some characterizations of rings in which every supplement of a finitely generated projective module is a direct summand, to those in which every supplement of a (non-necessarily finitely generated) projective module is a direct summand.Throughout this paper, R will denote an associative ring with 1, module will mean left R-module and morphisms will operate on the right. J (R) will denote the Jacobson radical of R and R-Mod will be the category whose objects are left R-modules. Recall that a submodule K of a module M is said to be superfluous (written K M) if for everyIt is trivial that a submodule K of a module M is a supplement of K M if and only if K + K = M and K ∩ K K. For every module M, J (M) will denote its Jacobson radical and given x ∈ R, (x : 0) will be the left ideal of R {r ∈ R: rx = 0}.

A study on dual square free modules

“…Suppose that U is a uniserial module. Then the endomorphism ring E = End R (U ) has two (two sided) ideals L = {f ∈ E : f is not injective} and K = {f ∈ E : f is not surjective} such that every proper right ideal of E is contained either in L or in K (see [11,Proposition 3.7] and [7,Theorem 1.2]). Now we give the following example which is important in terms of existing of an H-supplemented module that does not satisfy FIEP.…”

On image summand coinvariant modules and kernel summand invariant modules

“…A is said to be radical B -projective if, for any module X , any homomorphism f : A → X , and any epimorphism g : B → X , there exists a homomorphism h : A → B such that Im (f − gh) ≪ X (cf. [11,16]), and equivalently, for any epimorphism g : B → X and any homomorphism f : A → X , there exist a small epimorphism ρ : X → Y for some module Y , a homomorphism h : A → B such that ρgh = ρf ([17, Proposition 1.2]). A is said to be im-summand (im-coclosed, im-small, resp.)…”

A Variation of Coretractable Modules

“…By [13,Theorem 1.6], there exist 0 = x ∈ R and r 0 ∈ R such that K = x R and x 2 r 0 = x. Then x(xr 0 − 1) = 0.…”