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...

If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). ...

In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study weakly prime ideals. We investigate rings in which all graded...

The target of this study is to observe some of the algebraic structures of a single valued neutrosophic set. So, we introduce the concept of a neutrosophic submodule of a given classical module and investigate some of the crucial properties and characterizations of the proposed concept.

Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like submodules $Q$ such that $M/Q$ is aprimeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to intro...

The blob algebra is a finite-dimensional quotient of the Hecke type B which almost always quasi-hereditary. We construct indecomposable tilting modules for over field characteristic 0 in doubly critical case. Every module maximal highest weight either projective or an extension simple by module. Moreover, every submodule weight. conclude that graded Weyl filtration multiplicities this case are ...

Let Δ be an abelian group. By considering the notion multiplication of Δ-graded modules (see [7]) over a commutative Δ-graded ring with unity, we introduce the notion of product of two Δ-graded submodules which we use to characterize the Δ-graded prime submodules of a multiplication Δ-graded module. Finally we proved a graded version of Nakayama lemma for multiplication Δ-graded modules.

Abstract Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull global dimensions, obtaining lower bounds to complement the upper obtained by Wang. In fact, we show that many common such rings obey a stronger property, which call triangularity, allows us also classical ...

(1.2) PV~V where p is defined by p(s, t) = (t, s), (s, tET). We shall assume throughout that T is arcwise connected. Let 77 be an associative and anticommutative graded if-algebra with unit 1, where ii is a field. We assume throughout that 77'= 0 if i<0, and 77° = 7C-1. Let 77+ denote the submodule spanned by the elements of positive degree. 77 is a Hopf algebra over K if there is an algebra ho...