Artinian Modules Research Articles

Artinian* Modules

In this paper, we introduce and study Artinian* modules as a generalization of Artinian modules. We transfer several results of Artinian modules to Artinian* modules. We also provide several characterizations of this new class of modules. Furthermore, we investigate the existence of secondary representation for Artinian* modules over commutative regular rings. Finally, we characterize Artinian* modules in the amalgamated module construction.

Ascent and descent of Artinian module structures under flat base changes

Let φ : R → S be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an Artinian R-module A, the structure of A ⊗ R S is described. As an application, the Artinianess of certain local cohomology modules is clarified.

Fully Fuzzy Visible Modules With Other Related Concepts

In previous our research, the concepts of visible submodules and fully visible modules were introduced, and then these two concepts were fuzzified to fuzzy visible submodules and fully fuzzy. The main goal of this paper is to study the relationships between fully fuzzy visible modules and some types of fuzzy modules such as semiprime, prime, quasi, divisible, F-regular, quasi injective, and duo fuzzy modules, where under certain conditions it has been proven that each fully fuzzy visible module is fuzzy duo. In addition, there are many various properties and important results obtained through this research, which have been illustrated. Also, fuzzy Artinian modules and fuzzy fully stable modules have been introduced, and we study the relationships between these kinds of modules and fully fuzzy visible modules. Many other intersecting results we found.

A New View on Length Module

This paper consists of several new results about Length property of the module M. F-length of any Module comes from several concepts like Neotherian module and Artinian modules (Neo and Art) with comp- sition series. We proved that any Neo -module has submodule T and Neo -module is F-length. This implies that T also has F-length Property. Finally, some remarks, examples and definitions have been presented in this paper.

A note on a Cohen-type theorem for Artinian modules

A note on a Cohen-type theorem for Artinian modules

On images of subshifts under embeddings of symbolic varieties

AbstractWe show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule $\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism $\tau \colon \! \Sigma \to B^G$ , a subshift $\Delta \subset \Sigma $ is of finite type, respectively sofic, if and only if so is the image $\tau (\Delta )$ . Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.

FINITENESS PROPERTIES OF FORMAL LOCAL COHOMOLOGY MODULES

In this paper, we investigate some properties of top formal localcohomology FdimM=aMa (M). Among other things, we determine AttR(FdimM=aMa (M)),in the case that FdimM=aMa (M) is an artinian module. Also we show that FdimM=aMa (M)is artinian if and only if it is minimax..

On modules satisfying the descending chain condition on r-submodules

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we introduce the concept of r-Artinian modules which is a new generalization of Artinian modules. An R-module M is called an r-Artinian module if M satisfies the descending chain condition on r-submodules. Also, we call the ring R to be an r-Artinian ring if R is an r-Artinian R-module. We prove that an R-module M is an r-Artinian module if and only if its total quotient module is an Artinian module. In particular, we observe that r-Artinian modules generalize S-Artinian modules, for some particular multiplicatively closed subsets S of R. Also, we extend many properties of Artinian modules to r-Artinian modules. Finally, we use the idealization construction to give non-trivial examples of r-Artinian rings that are not Artinian.

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 strongly dccr⋆ modules

In this paper, we introduce and study the concept of strongly dccr[Formula: see text] modules. Strongly dccr[Formula: see text] condition generalizes the class of Artinian modules and it is stronger than dccr[Formula: see text] condition. Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] a unital [Formula: see text]-module. A module [Formula: see text] is said to be strongly [Formula: see text] if for every submodule [Formula: see text] of [Formula: see text] and every sequence of elements [Formula: see text] of [Formula: see text], the descending chain of submodules [Formula: see text] of [Formula: see text] is stationary. We give many examples and properties of strongly dccr[Formula: see text]. Moreover, we characterize strongly dccr[Formula: see text] in terms of some known class of rings and modules, for instance in perfect rings, strongly special modules and principally cogenerately modules. Finally, we give a version of Union Theorem and Nakayama’s Lemma in light of strongly dccr[Formula: see text] concept.

An explicit version of the correspondence between Noetherian modules with Cartier maps and Artinian modules with Frobenius maps

In this paper, we introduce an explicit version of the correspondence between Noetherian modules with Cartier maps and Artinian modules with Frobenius maps which recovers the correspondence given by R. Y. Sharp and Y. Yoshino, and the correspondence given by M. Blickle and G. Böckle.**********************************************************************************************************************************************************************************************************************************

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

A Cohen-type theorem for w-Artinian modules

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

On M-coidempotent elements and fully coidempotent modules

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.

The nice m-system of parameters for Artinian modules

This paper restates the definition of the nice m-system of parameters for Artinian modules. It also shows its effects on the differences between lengths and multiplicities of certain systems of parameters for Artinian modules: In particular, if is a nice m-system of parameters then the function is a polynomial having very nice form. Moreover, we will prove some properties of the nice m-system of parameters for Artinian modules. Especially, its effect on the annihilation of local homology modules of Artinian module A.

Some properties of intersection graph of a module with an application of the graph of ℤn

Let R be an unital ring which is not necessarily commutative. The intersection graph of ideals of R is a graph with the vertex set which contains proper ideals of R and distinct two vertices I and J are adjacent if and only if I Ç J ¹ 0 is denoted by G. In this paper, we will give some properties of regular graph, triangle-free graph and clique number of G(M) for a module M. We also characterize girth of an Artinian module with connected module. We characterize the chromatic number of G(Z n ). We also give an algorithm for the chromatic number of G(Z n ).

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

Reduction of Ideals Relative to an Artinian Module and the Dual of Burch’s Inequality

Let (A, 𝔪) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with ℓ(0 :M I) < ∞, then we show that for a minimal reduction J of I, (0 :M JI) = (0 :M I2) if and only if [Formula: see text] for all n ≥ 0. Moreover, we study the dual of Burch’s inequality. In particular, the Burch’s inequality becomes an equality if G(I, M) is co-Cohen-Macaulay.

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

On the dual of Hilbert coefficients and width of the associated graded modules over Artinian modules

Let (A,m) be a commutative quasi-local ring with non-zero identity and let M be an Artinian co-Cohen-Macaulay R-module with NdimM = d. Let I ? m be an ideal of R with ?(0:M I) < ?. In this paper, for 0 ? i ? d, we study the dual of Hilbert coefficients ?i(I,M) of I relative to M. Also, we prove the dual of Huckaba-Marley?s inequality. Moreover, we obtain some consequences of this result.