Arbitrary Direct Sums Research Articles

Direct sum of various generalizations of h-lifting modules

In this work, [Formula: see text]-lifting modules, a special extension of [Formula: see text]-lifting which is a further generalization of lifting modules, are studied. Here, we observed that finite direct sum of [Formula: see text]-lifting modules may not be [Formula: see text]-lifting, so we provided various sufficient conditions for which [Formula: see text]-lifting modules are closed under direct sum. Moreover, we have introduced and studied the properties of a new version of [Formula: see text]-lifting module namely, [Formula: see text]-lifting, [Formula: see text]-lifting and completely [Formula: see text]-lifting modules and develop some more properties of the [Formula: see text]-lifting modules in terms of these modules. Further, we proved that if the direct sum of arbitrary family of hollow modules is [Formula: see text]-lifting, then the arbitrary direct sum of [Formula: see text]-lifting [Formula: see text]-module is again [Formula: see text]-lifting.

A generalization of Gorenstein injective modules

Let R be an associative ring with identity and M be a left R-module. In this paper, we introduce M-Gorenstein injective modules as a generalization of Gorenstein injective modules. We verify some properties of M-Gorenstein injective modules analogous to those holding for Gorenstein injective modules. There is an interesting theorem in classical homological algebra which asserts that R is a Noetherian ring if and only if the class of injective modules over R is closed under arbitrary direct sum. Our goal in this paper is to investigate the M-Gorenstein injective counterpart of this fact. If the class of M-Gorenstein injective modules over R is closed under arbitrary direct sum, then R will be a Noetherian ring. Also, it has been proved that in the special case M=R, when R is a commutative Noetherian ring with a dualizing complex, then the class of R-Gorenstein injective modules is closed under arbitrary direct sum. In the main theorem of this paper, we prove the general case of this result. More precisely, we show that for any left R-module M over a Noetherian ring R in which every R-module has finite M-Gorenstein injective dimension, the class of M-Gorenstein injective modules is closed under arbitrary direct sum.

Covering modules by proper submodules

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover R-modules by proper submodules for commutative rings R, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number possibly infinite, such that a given R-module is a union of -many proper submodules. (1) We completely characterize when is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.

Quasi-s.Baer and related modules

In this paper, the s.Baer module concept and some of its generalizations (e.g. quasi-s.Baer, [Formula: see text]-s.Baer and p.q.-s.Baer) are developed. To this end, we characterize the class of rings for which every module is quasi-s.Baer as the class of rings which are finite direct sums of simple rings. Connections are made between the s.Baer (quasi-s.Baer, [Formula: see text]-s.Baer) and the extending (FI-extending, [Formula: see text]-extending) properties. We introduce the notions of quasi-nonsingularity (FI-s.nonsingular, [Formula: see text]-s.nonsingular) and [Formula: see text]-cononsingular (FI-[Formula: see text]-cononsingular, [Formula: see text]-[Formula: see text]-cononsingular) to extend the Chatters–Khuri theorem from rings to modules satisfying s.Baer or related conditions. Moreover, we investigate the transfer of various Baer properties between a module and its ring of scalars. Conditions are found for which some classes of quasi-s.Baer modules coincide with some classes of p.q.-s.Baer modules. Further we show that the class of quasi-s.Baer (p.q.-s.Baer) modules is closed with respect to submodules, extensions, and finite (arbitrary) direct sums. Examples illustrate and delimit our results.

On a class of dual Rickart modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

Injective DG-modules over non-positive DG-rings

Let A be an associative non-positive differential graded ring. In this paper we make a detailed study of a category Inj(A) of left DG-modules over A which generalizes the category of injective modules over a ring. We give many characterizations of this category, generalizing the theory of injective modules, and prove a derived version of the Bass–Papp theorem: the category Inj(A) is closed in the derived category D(A) under arbitrary direct sums if and only if the ring H0(A) is left noetherian and for every i<0 the left H0(A)-module Hi(A) is finitely generated. Specializing further to the case of commutative noetherian DG-rings, we generalize the Matlis structure theory of injectives to this context. As an application, we obtain a concrete version of Grothendieck's local duality theorem over commutative noetherian local DG-rings.

Direct sums of quasi-Baer modules

While the research work on quasi-Baer rings has been quite extensive in existing literature, the study of quasi-Baer modules introduced in 2004 remains quite limited with only little work available on the notion. In this paper, we extend the study of quasi-Baer modules. A characterization of a quasi-Baer module in terms of its endomorphism ring is obtained. We provide a complete characterization of arbitrary direct sums of quasi-Baer modules to be quasi-Baer. We also show characterizations of some important properties of quasi-Baer modules. Examples which delineate the concepts and the results are provided.

Smooth representations and Hecke modules in characteristicp

Let G be a p-adic Lie group and I ⊆ G be a compact open subgroup which is a torsionfree pro-p-group. Working over a coefficient field k of characteristic p we introduce a differential graded Hecke algebra for the pair (G, I) and show that the derived category of smooth representations of G in k-vector spaces is naturally equivalent to the derived category of differential graded modules over this Hecke DGA. 1 Background and motivation Let G be a d-dimensional p-adic Lie group, and let k be any field. We denote by Modk(G) the category of smooth G-representations in k-vector spaces. It obviously has arbitrary direct sums. We fix a compact open subgroup I ⊆ G. In Modk(G) we then have the representation indI (1) := all k-valued functions with finite support on G/I with G acting by left translations. Being generated by a single element, which is the characteristic function of the trivial coset, indI (1) is a compact object in Modk(G). It generates the full subcategory Modk(G) of all representations V in Modk(G) which are generated by their I-fixed vectors V I . In general Modk(G) is not an abelian category. The Hecke algebra of I by definition is the endomorphism ring HI := EndModk(G)(ind G I (1)) op . We let Mod(HI) denote the category of left unital HI -modules. There is the pair of adjoint functors H : Modk(G) −→ Mod(HI) V 7−→ V I = HomModk(G)(ind G I (1), V ) , and T0 : Mod(HI) −→ Modk(G) ⊆ Modk(G) M 7−→ indI (1)⊗HI M .

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

In this paper, we consider the lattice of fully invariant subgroups of the cotorsion hull of a separable p-group, which is assumed to be an arbitrary direct sum of torsion-complete p-groups. It is shown that this lattice is isomorphic to the lattice of filters of a lower semilattice made up of infinite matrices.

Extension groups between atoms and objects in locally noetherian Grothendieck category

We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some relationships to the theory of prime ideals in the case of noetherian algebras.

The homotopy category of injectives

Krause studied the homotopy category K(InjA) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K(InjA) has coproducts. It turns out that K(InjA) is compactly generated, and Krause studies the relation between the compact objects in K(InjA), the derived category D(A), and the category Kac(InjA) of acyclic objects in K(InjA). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K(InjA) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K(InjA) into K(A) has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D(A) is not compactly generated. I believe this is the first known example of such a thing.

$E_n$-regularity implies $E_{n-1}$-regularity

Vorst and Dayton-Weibel proved that K_n -regularity implies K_{n-1} -regularity. In this article we generalize this result from (commutative) rings to differential graded categories and from algebraic K -theory to any functor which is Morita invariant, continuous, and localizing. Moreover, we show that regularity is preserved under taking desuspensions, fibers of morphisms, direct factors, and arbitrary direct sums. As an application, we prove that the above implication also holds for schemes. Along the way, we extend Bass' fundamental theorem to this broader setting and establish a Nisnevich descent result which is of independent interest.

ℱ-Noetherian and Weakly Exact Torsion Theories

It is well-known (see [13]) that a hereditary torsion theory τ for the category R-mod is noetherian if and only if the class of all τ-torsionfree τ-injective modules is closed under arbitrary direct sums. So, it is natural to investigate the hereditary torsion theories having the property that the class of all τ-torsionfree injective modules is closed under arbitrary direct sums, which are called ℱ-noetherian. These torsion theories have been studied by Teply in [16]. In the second part of this note we shall study the weakly exact hereditary torsion theories, which generalize the exact one's.

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

The paper considers the lattice of fully invariant subgroups of the cotorsion hull when a separable primary group T is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.

Thick subcategories of modules over commutative noetherian rings (with an appendix by Srikanth Iyengar)

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In addition, subcategories of A-modules that are closed under taking submodules, extensions, and direct unions are classified via associated prime ideals.

Unions of slender groups

It is well-known that the class of slender groups is closed under extensions, arbitrary direct sums and subgroups. We show that it is also closed under taking unions of continuous well-ordered ascending chains of groups where each factor of consecutive groups is slender (Theorem 2.2). We also prove that the class of slender groups cannot be obtained from any set of slender groups by forming unions of chains of the mentioned kind and taking subgroups (Theorem 4.2). This generalizes a result by Gobel-Wald [11].

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called ⊕-cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the ⊕-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of ⊕-cofinitely supplemented modules is ⊕-cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is ⊕-cofinitely supplemented. In addition, if M has the summand sum property, then M is ⊕-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.

Baer and Quasi-Baer Modules#

We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.

Direct Sums of Quotient Divisible Groups

An abelian group G is quotient divisible (qd) if it is of finite torsion-free rank and there exists a free subgroup F ⊂ G with G/F a divisible torsion group. We study the category ∑𝒟 with objects arbitrary direct sums of qd groups and maps quasi-homomorphisms. First we identify the qd-indecomposable groups, that is the indecomposables in the category ∑𝒟. These turn out to be the qd groups G such that, if G ⊃ A ⊕ B ⊃ tG for t > 0, then A or B is finite. The major tool for studying ∑𝒟 is the notion of an admissable quasi-decomposition G ≈ ⊕ G k of a group G ∈ ∑𝒢 into a direct sum of qd-indecomposable subgroups. (See Definition 6 below.) For any qd-indecomposable group A and any admissable quasi-decomposition G ≈ ⊕ G k , the number of G k quasi-isomorphic to A is an invariant of G, denoted σ A (G). The cardinal numbers σ A (G), together with the finite Ulm invariants (G), form a complete set of isomorphism invariants for groups G = ⊕G i ∈ ∑𝒢 with p-rank[G i /T(G i )] ≤ 1 for all p, i. Let G ∈ ∑𝒟 with G = A ⊕ B. If G ≈ ⊕ G k , then there is an admissable quasi-decomposition A ≈ ⊕ A j such that each A j is quasi-isomorphic to some G k(j). We consider the problem: When is an abelian group A with an admissable quasi-decomposition A ≈ ⊕ A j into qd-indecomposable subgroups a direct summand of a group in ∑𝒟? We get a positive answer if A is torsion-free or if the mixed groups A j are all in a class 𝒢 ⊂ 𝒟.

Sublattices of the Lattice of Pre-natural Classes of Modules

A pre-natural class of modules over a ring R is one that is closed under isomorphic copies, submodules, arbitrary direct sums, and certain essential extensions. The set Npr(R) of all pre-natural classes is a lattice, which contains many previously studied lattices of R-module classes. The sublattice structure of Npr(R) is studied in this paper and is related to ring and module properties of R