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For a commutative, unital and integral quantale V, we generalize to V-groups the results developed by Gran and Michel for preordered groups. We first of all show that, in the category V-Grp of V-groups, there exists a torsion theory whose torsion and torsion-free subcategories are given by those of indiscrete and separated V-groups, respectively. It turns out that this torsion theory induces a monotone-light factorization system that we characterize, and it is then possible to describe the coverings in V-Grp. We next classify these coverings as internal actions of a Galois groupoid. Finally, we observe that the subcategory of separated V-groups is also a torsion-free subcategory for a pretorsion theory whose torsion subcategory is the one of symmetric V-groups. As recently proved by Clementino and Montoli, this latter category is actually not only coreflective, as it is the case for any torsion subcategory, but also reflective., Comment: 21 pages

Let R be a commutative ring with identity and τ be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of τ-comultiplication module. We present several propert...

We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s clas...

For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$--artinian ring, then $\sigma$ must be of finite type, and $A$ is totally $\sigma$--noetherian., Comment: 22 pages

In this article we explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli., 21 pages, minor changes and improvements

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when $\mathcal Z$ is reduced to the $0$-object of $\mathcal C$. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets., 22 pages

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.

In this paper we introduce and investigate the latticial counterpart of the module-theoretical concept of preradical. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

Using the concept of prime submodule for M ∈ R-Mod, P ∈ Spec(M), and N ∈ σ[M], we define when N is P-Mtame \ (Mtame) module. This concept generalizes the concept\ of P-tame (tame) modules. For M ∈ R-Mod and τ ∈M-tors, we use the concept of τ M -Gabriel dimension and we study the relationship between Mtame modules and τ M -Gabriel dimension. We find equivalent conditions for a module M progenerator in σ[M] with τ M -Gabriel dimension to have τ M -Gabriel correspondence in terms of the P-Mtame modules. This result extends the results by Albu et al. and Kim and Krause.

Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in $\mathfrak{Qcoh}(X)$, for $X$ an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi--coherent sheaves on a Dedekind scheme $X$ is closed under arbitrary direct products, and that the class of all locally torsion-free quasi--coherent sheaves induces a hereditary torsion theory on $\mathfrak{Qcoh}(X)$. Finally torsion-free covers are shown to exist in $\mathfrak{Qcoh}(X)$.

Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K tor × K d , where K tor is a bounded torsion compact abelian group and K d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9].

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

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 formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, {\it Int. Elect. J. Algebra}, {\bf 13}, (2013), 1--14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlited and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules, (i.e., modules whose all submodules are completely prime). For a fully completely prime left $R$-module $M$, if $a, b\in R$ and $m\in M$, then $abm=bam$, $am=a^km$ for all positive integers $k$, and either $am=abm$ or $bm=abm$. In the last section, two different torsion theories induced by the completely prime radical are given., Comment: 15 pages

We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent.

The aim of this article is to present a latticial version of the renown module theoretical Osofsky–Smith Theorem.

Let [Formula: see text]be a finite direct sum of modules. We prove: (i) If Mi is radical Mj-projective for all j > i and each Mi is H-supplemented, then M is H-supplemented. (ii) If all the Mi are relatively projective and N is H-supplemented, then each Mi is H-supplemented. Let ρ be the preradical for a cohereditary torsion theory. Let M be a module such that ρ (M) has a unique coclosure and every direct summand of ρ (M) has a coclosure in M. Then M is H-supplemented if and only if there exists a decomposition M=M1⊕M2 such that M2 ⊆ ρ(M), ρ(M)/M2 ≪ M/M2, and M1, M2 are H-supplemented.

Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and perfect cotorsion theory.

Let R be a ring. A right R-module U is called Tor-tilting if , where U + = Homℤ(U, ℚ/ℤ), Cogen(U +) is the class of left R-modules cogenerated by U + and consists of modules R M such that . Some examples and characterizations of Tor-tilting modules are given. Among others, it is shown that U R is Tor-tilting if and only if U + is cotilting. Moreover, both tilting modules and completely faithful flat modules are proved to be Tor-tilting. The torsion theory induced by a Tor-tilting module is also investigated.

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory. Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.

A Grothendieck category C is said to be locally finitely generated if the subobject lattice of every object in C is compactly generated, or equivalently, if C possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category C indecomposable if C is not equivalent to a product of nonzero Grothendieck categories C1×C2. In this paper an example of an indecomposable nonlocally finitely generated Grothendieck category possessing simple objects is constructed, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [T. Albu, M. Iosif, M.L. Teply, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) 52–79].

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.

We prove that every perfect torsion theory for a ring R is differential (in the sense of [P.E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1–8]). In this case, we construct the extension of a derivation of a right R -module M to a derivation of the module of quotients of M . Then, we prove that the Lambek and Goldie torsion theories for any R are differential.

Let MR be a right R-module over a ring R with S = End (MR). We study the coherence of the left S-module SM relative to a torsion theory for the category of right R-modules. Various results are developed, many extending known results.

Let M be a module and K a submodule of a module N in σ[M]. We call K a δ–M-small submodule of N if whenever N = K + L, N/L is M-singular for any submodule L of N, we have N = L. Also we call N a δ–M-small module if N is a δ–M-small submodule of its M-injective hull. In this article, we consider , the reject of  ℳ in N, where  ℳ is the class of all δ–M-small modules. We investigate the properties of and consider the torsion theory τδ V in σ[M] cogenerated by  ℳ. We compare the τδ V and the torsion theory τ V cogenerated by M-small modules and finally we give a characterization of GCO-modules.

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups. (c) 2005 Elsevier B.V. All rights reserved.

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie $K$-algebras, respectively.

application/pdf, We show that if A is a representation-finite selfinjective Artin algebra, then every P• ∈ Kb(PA) with HomK(Mod-A)(P•,P•[i]) = 0 for i ≠ 0 and add(P•) = add(νP•) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B0, B1,…, Bm = B such that, for any 0 ≤ i < m, Bi+1 is the endomorphism algebra of a tilting complex for Bi of length ≤ 1.

In this note we give a new and elementary proof of a result of Nastasescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra \(C^*\) is a direct summand in M (the splitting problem for coalgebras).

Let M R be a right R-module over a ring R with S = End(M R ). We study the coherence of the left S-module S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.

After recapitulating the rudiments of the Kurosh-Amitsur radical theory of S-acts, hereditary radicals are discussed. The hereditary Hoehnke radical assignments r which designate a Rees congruence r(A) to each S-act A, are the hereditary Kurosh-Amitsur radical assignments. Then the corresponding radical as well as semisimple classes are characterized. Equivalence classes of injective S-acts determine hereditary torsion assignments t, these are just the hereditary Hoehnke radicals, but the congruence t(A) of A need not be a Rees congruence. Torsion and torsionfree classes are characterized; several hereditary torsion assignments may determine the same torsion class which is always a radical class closed under taking subacts. Examples show that a hereditary torsion assignment need not be a hereditary radical.

Let M be a right R-module, \({\cal M}\) the class of all M-small modules, and P a projective cover of M in \(\sigma\)[M]. We consider the torsion theories \(\tau_{\cal M}\) = (\({\cal T}_{\cal M}, {\cal F}_{\cal M}\)), \(\tau_V\) = (\({\cal T}_V, {\cal F}_V\)), and \(\tau_P\) = (\({\cal T}_P, {\cal F}_P\)) in \(\sigma\)[M], where \(\tau_{\cal M}\) is the torsion theory generated by \({\cal M}, \tau_V\) is the torsion theory cogenerated by \({\cal M}\), and \(\tau_P\) is the dual Lambek torsion theory. We study some conditions for \(\tau_{\cal M}\) to be cohereditary, stable, or split, and prove that Rej(M, \({\cal M}\)) = M\(\Leftrightarrow\)\({\cal F}_P\) = \({\cal M}\) (= \({\cal T}_{\cal M}\) = \({\cal F}_V\)) \(\Leftrightarrow\)\({\cal T}_P\) = \({\cal T}_V\)\(\Leftrightarrow\) GenM(P) \(\subseteq\)\({\cal T}_V\).

Conditions under which a natural class of right R-modules is closed under quotient modules are determined. In this case the unique complementary natural class of c(Δ) is closed under direct products. Hence (Δ, c(Δ)) is a hereditary torsion theory with torsion class Δ. In general a natural class [Formula: see text] of right R-modules is not closed under direct products (or quotient modules), yet it has been shown by Y. Zhou that each natural class [Formula: see text] determines a unique hereditary torsion theory τ. The torsion and torsion free classes of this torsion theory τ are studied, and in particular, their dependence on the original natural class [Formula: see text]. As an application, the resulting torsion theories τ are used to define the class of τ-simple, i.e., τ-cocritical modules. Most of the above is done more generally for M-natural classes in the category σ[M].

Let S be a ring extension of R, and let τ R =( T R , F R ) and τ S =( T S , F S ) be torsion theories for right R-modules and right S-modules respectively. Using functors HomR(−,S) and −⊗RS, we get the connection between τR and τS, introduce torsion theory extensions related to ring extensions and establish some relations between finite normalizing extensions and torsion theory extensions. Furthermore, we consider some applications of torsion theory extensions and generalize some results of excellent extensions (almost excellent extensions) to the setting of torsion theories.

The concept of pseudo-regular S-system is introduced. A torsion theory \(\tau _u = \left( {U_S ,\bar U_S } \right)\) with its torsion class US consisting of pseudo-regular S-systems, is constructed. Its corresponding quasi-filter is completely described. A number of results on the relations between τu and two special torsion theories, the stable and Lambek torsion theories, are obtained.

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C-*, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C-*. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive. (C) 2002 Elsevier Science B.V. All rights reserved.

In this article, the conditions for a Rees short exact sequence of S -systems to be left and right split, respectively, are given. We note that this result differs from the well known result of an exact sequence of modules that is split.

Let M and N be R-modules. We define where S denotes the class of all M-small modules. We call N an M-cosingular (non-M-cosingular) module if Z M (N) = 0 ( Z M (N) = N). We study the properties of M-cosingular and non-M-cosingular modules in σ[M] We consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented. We also give instances where this torsion theory is cohereditary or splits. Finally we characterise lifting modules N ∊ [M] in terms of Z 2 M (N).

We characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathfrak{t}$ on a stable $\infty$-category $\mathbf{C}$ is equivalent to a normal torsion theory $\mathbb{F}$ on $\mathbf{C}$, i.e. to a factorization system $\mathbb{F}=(\mathcal{E},\mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts., Minor typographical corrections from v1; 25 pages; to appear in "Applied Categorical Structures"

Cotilting bimodules over arbitrary rings give rise to a theory which naturally generalizes Morita dualities in the setting of torsion theory. Here we study the case when the torsion theories cogenerated by a cotilting bimodule are hereditary.

Necessary and sufficient conditions are provided for the nilpotence of the intersection of all τ-closed prime ideals of a ring R with τ-Krull dimension, where τ denotes a hereditary torsion theory on the category Mod-R of right R-modules. It is shown that these conditions hold, in particular, when τ is ideal invariant. Furthermore, conditions are provided for Ass(M) to be non-empty for every τ-torsionfree right R-module M ≠ 0.

Let α and β be automorphisms on a ring R and δ : R → R an ( α , β ) -derivation. It is shown that if F is a right Gabriel filter on R then F is δ -invariant if it is both α and β -invariant. A consequence of this result is that every hereditary torsion theory on the category of right R -modules is differential in the sense of Bland (2006). This answers in the affirmative a question posed by Vas (2007) and strengthens a result due to Golan (1981) on the extendability of a derivation map from a module to its module of quotients at a hereditary torsion theory.

Conditions under which a *-module AU extends to a *-module RR⊗AU over a ring extension R of A are presented, along with con¬nections to quasi-progenerators, tilting modules, and torsion theory counter equivalences.

Contravariantly finite subcategories have been useful in differentaspects of representation theory (Auslander and Reiten, 1989, 1991;Auslander and Smalo, 1981) and they appear very naturally, forexample, the torsion class of a torsion theory is contravariantly finite. Inthis paper we explore further relations between contravariantly finite,resolving, subcategories and torsion theories. We study these connections inthe category of functors that vanishes on projectives. Our resultsgeneralize some theorems we obtained previously in our interpretation ofNakayama’s conjecture (Martinez-Villa, 1994).

This article introduces the concept of μ-complemented modules as follows: given a hereditary torsion theory in Mod-R with associated torsion functor μ we say that a module M is μ-complemented when every sub module is μ- dense in a summand of M. We present here some fundamental properties of this class of modules and study the relationship between them and extending modules. We apply these results, in particular, when μ is the torsion functor associated to the Goldie torsion theory and when μ is jansian. The question of which direct sums of μ-complemented R-modules are μ-complemented is treated here. We also obtain decomposition theorems for μ-complemented modules when either the ring or the module satisfies certain chain conditions.

Let M be an R-module and α[M] the category of all R-modules that are subgenerated by M. For any hereditary torsion theory T in α[M], the class of quotient modules defines a Grothendieck category denoted by ∊T[M]. T is called spectral if ∊T[M] is a spectral category (every short exact sequence in∊T[M] splits). The paper is devoted to the investigation of such torsion theories. Various characterizations are obtained and properties of the endomorphism rings of the corresponding quotient modules are studied. The final sectlon is concerned with the role of simple and semisimple modules with respect to spectral torsion theories. Our presentation generalizes and extends related results for spectral torsion theories in the category of all R-modules R-Mod.

In a previous paper we introduced μ-complemented modules for any given torsion functor μ of a hereditary torsion theory in Mod-R and presented their basic properties. Our current research enables us to characterize these modules when μ is a proper torsion theory other Than Goldie's and the ring is a one-dimensional Noetherian domain. The case in which μ has property (T) is also treated. For an arbitrary μ, rings for which every module is μ-complemented are considered.

Torsion classes and their corresponding quasi-filters in S-systems will be investigated. Torsion theory, in particular the hereditary torsion theory of S-systems will be characterized by the conguences defined on a semigroup S. A number of results on torsion theory of semigroups in the literature will be generalized and extended.

Let [Formula: see text] be an arbitrary ring with identity and [Formula: see text] a right [Formula: see text]-module with the ring [Formula: see text] End[Formula: see text] of endomorphisms of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text], where [Formula: see text] is a fully invariant submodule of [Formula: see text], is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule [Formula: see text] of [Formula: see text] as [Formula: see text] and [Formula: see text], and investigate some properties of [Formula: see text]-inverse split modules and [Formula: see text]-inverse split modules [Formula: see text]. Results are applied to characterize rings [Formula: see text] for which every free (projective) right [Formula: see text]-module [Formula: see text] is [Formula: see text]-inverse split for the preradicals such as [Formula: see text] and [Formula: see text].

(1995). Tilting and torsion theory counter equivalences. Communications in Algebra: Vol. 23, No. 13, pp. 4833-4849.