The main goal of this article is to explore the concepts graded ϕ-2-absorbing and primary submodules as a new generalization 2-absorbing submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be function, where GS(M) denotes collection R-submodules M. A proper K∈GS(M) said R-submodule M if whenever x,y are homogeneous elements R s element with xys∈K−ϕ(K), then xs∈K or ys∈K xy∈(K:RM), we call K xs ys in radical xy∈(...

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of classical strongly 2-absorbing second submodules modules over rings. A number results concerning these classes their homogeneous components are given.

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a di...

The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

Let R be a commutative ring with nonzero identity and M an R-module. In this paper, first we give some relations between S-prime S-maximal submodules that are generalizations of prime maximal submodules, respectively. Then construct topology on the set all , which is generalization spectrum M. We investigate when SpecS(M) T0 T1-space. also study continuous maps irreducibility SpecS(M). Moreover...

We extend the “linearly exponential” bound for the Castelnuovo-Mumford regularity of a graded ideal in a polynomial ring K[x1, . . . , xr] over a field (established by Galligo and Giusti in characteristic 0 and recently, by Caviglia-Sbarra for abitrary K) to graded submodules of a graded module over a homogeneous Cohen-Macaulay ring R = ⊕n≥0Rn with artinian local base ring R0. As an application...