Artinian Modules Research Articles

On Perfect Commutative EIFA-rings

We consider the class F of endo-Artinian modules, i.e. the modules M which satisfying the descending chain condition for endomorphic images: any descending chain Imf1 ⊃ Imf2 ⊃ · · · ⊃ Imfn ⊃ · · · is stationary, where fi ∈ End(M). Let A be the class of Artinian modules. It is clear that every Artinian R-module M is endo-Artinian, soA ⊂ F , but the converse is not true. Indeed, Q is a non-Artinian Z-module which is endo-Artinian. The aim of this work, is to characterize perfect commutative rings for which F and A are identical.

ON DIRECT SUM DECOMPOSITIONS OF KRULL–SCHMIDT ARTINIAN MODULES

We study direct sum decompositions of modules satisfying the descending chain condition on direct summands. We call modules satisfying this condition Krull–Schmidt artinian. We prove that all direct sum decompositions of Krull–Schmidt artinian modules refine into finite indecomposable direct sum decompositions and we prove that this condition is strictly stronger than the condition of a module admitting finite indecomposable direct sum decompositions. We also study the problem of existence and uniqueness of direct sum decompositions of Krull–Schmidt artinian modules in terms of given classes of modules. We present also brief studies of direct sum decompositions of modules with deviation on direct summands and of modules with finite Krull–Schmidt length.

Annihilators of Artinian modules compatible with a Frobenius map

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field.The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the F-finite case.

A FINITENESS RESULT FOR ATTACHED PRIMES OF ARTINIAN LOCAL COHOMOLOGY MODULES

Let (R, 𝔪) be a Noetherian local ring. For an integer s ≥ -1 and an Artinian R-module A, we introduce the notion of A-cosequence in dimension > s and show that the set of all attached primes [Formula: see text] satisfying dim (R/𝔭) ≥ s is a finite set whenever (x1, …, xk) is an A-cosequence in dimension > s. As an application, we give a finiteness result for attached primes of certain Artinian local cohomology modules of a finitely generated R-module.

RELATIVE ATTACHED PRIMES AND COREGULAR SEQUENCES

We extend the existing concepts of secondary representation of a module, coregular sequence and attached prime ideals to the more general setting of any hereditary torsion theory. We prove that any $\tau$-artinian module is $\tau$-representable and that such a representation has some sort of unicity in terms of the set of $\tau$-attached prime ideals associated to it. Then we use $\tau$-coregular sequences to find a nice way to compute the relative width of a module. Finally we give some connections with the relative local homology.

Second Modules Over Noncommutative Rings

Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P i (1 ≤ i ≤ k) such that if L is a proper submodule of M with M/L a second module, then M/L has annihilator P i for some 1 ≤ i ≤ k. Every second submodule of an Artinian module is a finite sum of hollow second submodules.

On modules which satisfy the radical formula

In this paper, the authors prove that every representable module over a commutative ring with identity satisfies the radical formula. With this result, they extend the class of modules satisfying the radical formula from that of Artinian modules to a larger one. They conclude their work by giving a description of the radical of a submodule of a representable module.

Some results on local homology and local cohomology modules

In this paper, we obtain some results about the local homology modules of Artinian modules, and by Matlis duality we obtain some results about the local cohomology modules of finitely generated modules.

Generalized Local Homology for Artinian Modules

We introduce generalized local homology which is in some sense dual to generalized local cohomology, and study some properties of generalized local homology modules for artinian modules, such as the artinianness, noetherianness and the characterization of Width I(M) by generalized local homology. By using duality, we get back some properties of generalized local cohomology.

A Finiteness Result for Attached Primes of Certain Tor-Modules

In this paper we introduce the notion of width in dimension > s for Artinian modules and give a finiteness result for attached primes of certain Tor-modules.

Direct-sum decompositions of modules with semilocal endomorphism rings

According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.

Remarks on subcategories of Artinian modules

We study two subcategories of the category of Artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of Artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring. These results are Artinian analogues of the theorems proved by Takahashi.

ATTACHED PRIMES UNDER SKEW POLYNOMIAL EXTENSIONS

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra212 (2008) 510–521], a theory of attached prime ideals in noncommutative rings was developed as a natural generalization of the classical notions of attached primes and secondary representations that were first introduced in 1973 as a dual theory to the associated primes and primary decomposition in commutative algebra (see [I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math.11 (1973) 23–43]). Associated primes over noncommutative rings have been thoroughly studied and developed for a variety of applications, including skew polynomial rings: see [S. A. Annin, Associated primes over skew polynomial rings, Commun. Algebra30(5) (2002) 2511–2528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl.3(2) (2004) 193–205]. Motivated by this background, the present article addresses the behavior of the attached prime ideals of inverse polynomial modules over skew polynomial rings. The goal is to determine the attached primes of an inverse polynomial module M[x-1] over a skew polynomial ring R[x;σ] in terms of the attached primes of the base module MR. This study was completed in the commutative setting for the class of representable modules in [L. Melkersson, Content and inverse polynomials on artinian modules, Commun. Algebra26(4) (1998) 1141–1145], and the generalization to noncommutative rings turns out to be quite non-trivial in that one must either work with a Bass module MR or a right perfect ring R in order to achieve the desired statement even when no twist is present in the polynomial ring "Let MR be a module over any ring R. If M[x-1]R is a completely σ-compatible Bass module, then Att (M[x-1]S) = {𝔭[x] : 𝔭 ∈ Att (MR)}." The sharpness of the results are illustrated through the use of several illuminating examples.

The Relationships between Relatively Cancellation Modules and Certain Types of Modules

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

Modules in σ[M] with Chain Conditions on δ<sub>M</sub>-small Submodules

Problem statement: Let M be a right module over a ring R. In this article modules in σ[M] with chain conditions on δM- small submodules are studied. Approach: With the help of known results about M- singular, Artinian and Noetherian modules the techniques of the proofs of our main results use the properties of δM- small, δM- supplement and δM- semimaximal submodules. Results: Modules in σ[M] with chain conditions on δM- small are investigated, δM- semimaximal submodule is defined . Some Properties ofδM- semimaximal submodules are proved. As application a new characterization of Artinian module in σ[M] is obtained in terms of δM- small submodules and δM- semimaximal submodules, as well as δM- small submodules and δM- supplement submodules. Conclusion/Recommendations: Our results certainly generalized several results obtained earlier.

Homological properties of noncommutative Iwasawa algebras

For any compact p-adic Lie group G, the Iwasawa algebra Ω G is an Artin–Schelter Gorenstein algebra. We obtain the Auslander–Buchsbaum formula, the Bass's theorem and the No-holes theorem for noetherian modules over Λ G and Ω G , and the dual versions for their artinian modules. It is shown that Ω G is Morita self-dual via dualizing complexes. We finally consider the homological invariant “grade” of filtered modules over Λ G and Ω G , when G is a uniform pro- p group with certain properties.

On the Catenaricity of Noetherian Local Rings and Quasi Unmixed Artinian Modules

Let (R, 𝔪) be a Noetherian local ring, A an Artinian R-module, and M a finitely generated R-module. Consider the following property: We say that A is quasi unmixed if for all In this article, we give a connection among the property (*) for a quasi unmixed Artinian module A, the catenaricity of the ring R/Ann R A, and the dimension of A. Specially, we show that for an integer i ≥ 0, if the local cohomology module is quasi unmixed then satisfies the property (*) if and only if the ring is catenary and

On Artinian Modules Over Duo Rings

The well-known method, which reduces the study of Artinian modules over commutative rings to the study of Artinian modules over quasi-local rings, is extended to the study of these modules over duo rings. The duals of Krull Intersection Theorem and Nakayama's Lemma are proved for Artinian modules over a large class of duo rings. We also give an upper bound for the length of a duo ring R in terms of the length of a faithful R-module. This generalizes a well-known result of Schur concerning the cardinality of a maximal linearly independent set of commuting matrices.

Generalized Weyl algebras: Category O and graded Morita equivalence

We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalence between two GWAs. We define a new algebra related to GWAs, and use it to produce some strongly graded Morita equivalences. Finally, we give a complete answer to the strongly graded Morita problem for classical GWAs.

A non-finitely generated algebra of Frobenius maps

This paper describes an Artinian module over a ring of prime characteristic whose algebra of Frobenius maps is not finitely generated. This settles a question raised by Lyubeznik and Smith.