Some Results and Examples on The Relation of Leray Equivalent and (Strong) Shift Equivalent

Using the concept of Leray functor and the way it is used to define the Cohomological Conley index, we define Leray equivalent, Leray shift equivalent, Leray elementary strong shift equivalent and Leray strong shift equivalent. We established their relationship with shift equivalent, strong shift equivalent and elementary strong shift equivalent. Then, we show that in the setting of Artinian and Noetherian module Leray equivalent implies strong shift equivalent.

On the second spectrum and the second classical Zariski topology of a module

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected.

Rings Graded By a Generalized Group

The aim of this paper is to consider the ringswhich can be graded by completely simple semigroups. We show that each G-graded ring has an orthonormal basis, where G is a completely simple semigroup. We prove that if I is a complete homogeneous ideal of a G-graded ring R, then R/I is a G-graded ring.We deduce a characterization of the maximal ideals of a G-graded ring which are homogeneous. We also prove that if R is a Noetherian graded ring, then each summand of it is also a Noetherian module..

Spaces with Noetherian cohomology

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

Rees Modules and Reduction of an Ideal Relative to a Noetherian Module

Let (A, 𝔪) be a Noetherian local ring, E a non-zero finitely generated A-module, and 𝔟 a proper ideal of A. The aim of this paper is, under some assumptions on𝔟 and E, to clarify a necessary and sufficient condition for the Rees module of E associated to 𝔟 to obtain Cohen-Macaulayness. Also, among other independent results about the reduction number of 𝔟 relative to E, we expand a theorem of Marley.

RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN

R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

Generalized lifting modules

We introduce the concepts of lifting modules and (quasi-)discrete modules relative to a given left module. We also introduce the notion of SSRS-modules. It is shown that (1) ifMis an amply supplemented module and0→N′→N→N″→0an exact sequence, thenMisN-lifting if and only if it isN′-lifting andN″-lifting; (2) ifMis a Noetherian module, thenMis lifting if and only ifMisR-lifting if and only ifMis an amply supplemented SSRS-module; and (3) letMbe an amply supplemented SSRS-module such thatRad(M)is finitely generated, thenM=K⊕K′, whereKis a radical module andK′is a lifting module.