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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. [ABSTRACT FROM AUTHOR]

We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold. [ABSTRACT FROM AUTHOR]

Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow loops are explicitly determined. Moreover, in contrast to what is the case for the well-known algebraic categories, it is shown here that such radicals for graphs need not be Kurosh–Amitsur radicals. [ABSTRACT FROM AUTHOR]

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 classification of Serre subcategories. [ABSTRACT FROM AUTHOR]

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions ( C) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions ( C) or ( C). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory. [ABSTRACT FROM AUTHOR]

We first characterize τ-complemented modules with relative (pre)covers. We also introduce an extending module relative to τ-pure submodules on a hereditary torsion theory τ and give its relationship with τ-complemented modules. [ABSTRACT FROM AUTHOR]

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show thatis a coalgebra such that the rational part of every left finitely generated-module splits off if and only ifis an upper triangular matrix coalgebra, for a serial coalgebrawhose Ext-quiver is a finite union of cycles, a finite dimensional coalgebraand a finite dimensional bicomodule. [ABSTRACT FROM AUTHOR]

Using the nice properties of thew-divisible weight and thew-divisible groups, we prove a factorization theorem for compact abelian groupsK; namely,K = Ktor × Kd, whereKtoris a bounded torsion compact abelian group andKdis aw-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9]. [ABSTRACT FROM AUTHOR]

This is the second part of the paper with the same title published inCommunications in Algebrain 2011. It contains applications of the Latticial Osofsky–Smith Theorem to Grothendieck categories and module categories equipped with a torsion theory. Various many different meanings spread in the literature of the relative concepts with respect to a hereditary torsion theory τ on Mod-Rlike τ-essential submodule, τ-complement submodule, τ-CS module, etc. are also discussed. [ABSTRACT FROM PUBLISHER]

Abstract: Given a torsion theory in an abelian category , the reflector to the torsion-free subcategory induces a reflective factorisation system on . It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Paré that induces a monotone-light factorisation system by simultaneously stabilising and localising , whenever the torsion theory is hereditary and any object in is a quotient of an object in . We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules. [Copyright &y& Elsevier]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Let R be a ring. A collection of R-modules containing the zero module and closed under isomorphisms will be denoted by X. An R-module M is said to be X -lifting if for every X-submodule N of M there exists A ≤ N such that M = A ⊕ B and N ∩ B is small in B [11]. In the present paper, we consider the question: Can we characterize X-lifting modules via objects of the class X ? [ABSTRACT FROM AUTHOR]

Let R be a ring and let A be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by A into the derived category of R has a right adjoint, denoted CellA. Recently, Benson has shown how to compute CellA R when R is a group ring of a finite group over a prime field and A is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where A is any hereditary torsion class on R. It is shown that for every R-module M there exists an injective R-module E such that: Hn (CellAM) ≅ Ext Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (Hom R(M,E),E) for n ≥ 2. [ABSTRACT FROM AUTHOR]

An R-module M is said to be a C1-module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of the τ-C1 module for a hereditary torsion theory τ on Mod-R. τ -C1 modules are a generalization of C1-modules. [ABSTRACT FROM AUTHOR]

Abstract: A Grothendieck category is said to be locally finitely generated if the subobject lattice of every object in is compactly generated, or equivalently, if possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category indecomposable if is not equivalent to a product of nonzero Grothendieck categories . 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]. [Copyright &y& Elsevier]

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients. [ABSTRACT FROM AUTHOR]

Abstract: Let A be a pseudocompact (or profinite) algebra, so where C is a coalgebra. We show that the if the semiartinian part (the “Dickson” part) of every A-module M splits off in M, then A is semiartinian, giving thus a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith. [Copyright &y& Elsevier]

Given a hereditary torsion theory τ = (T,F) in Mod-R, a module M is called τ-supplemented if every submodule A of M contains a direct summand C of M with A/C τ-torsion. A submodule V of M is called τ - supplement of U in M if U + V = M and U ∩ V ≤(V) and M is τ - weakly supplemented if every submodule of M has a τ-supplement in M. Let M be a τ-weakly supplemented module. Then M has a decomposition M = M1 ⊕ M2 where M1 is a semisimple module and M2 is a module with τ(M2) ≤e M2. Also, it is shown that; any finite sum of τ-weakly supplemented modules is a τ-weakly supplemented module. [ABSTRACT FROM AUTHOR]

Abstract: In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category are introduced and are shown to yield a version of the classical tilting theorem between and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory in , for a ring R, the corresponding Heart is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and . In this paper we first prove that is a prototype for any abelian category admitting a tilting object which tilts to in . Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that is a Grothendieck or even a module category. As particular situations, we examine two main cases: when is hereditary cotilting, proving that is Grothendieck and when is tilting, proving that is a module category. [Copyright &y& Elsevier]

We show that if A is a representation-finite selfinjective Artin algebra, then every P• ∈ Kb(&Pscr;A) 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. [ABSTRACT FROM AUTHOR]

Abstract: Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class . First, we show that a finitely generated module over a ring from the class splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from and prove that this dimension has all the nice properties of the dimension studied in [W. Lück, J. Reine Angew. Math. 495 (1998) 135–162] for finite von Neumann algebras. This dimension defines a torsion theory that we prove to be equal to the Goldie and Lambek torsion theories. Moreover, every finitely generated module splits in this torsion theory. If R is a ring in , we can embed it in a canonical way into a regular ring Q also in . We show that is isomorphic to by producing an explicit isomorphism and its inverse of monoids that extends to the isomorphism of and . [Copyright &y& Elsevier]

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules. [ABSTRACT FROM AUTHOR]

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple? [ABSTRACT FROM AUTHOR]

This article introduces the concept of a τ-supplemented module as follows: Given a hereditary torsion theory in Mod R with associated torsion functor τ, we say that a module M is τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. We present here some fundamental properties of this class of modules and study the decompositions of τ-supplemented modules under certain conditions on modules. The question of which direct sum of τ-supplemented R-modules are τ-supplemented is treated here. [ABSTRACT FROM AUTHOR]

Let R and S be two rings. Each category equivalence between a torsion class of left (right) R-modules and a torsion-free class of left (right) S-modules is represented by a left (right) quasitilting triple. Suppose we have a pair of equivalences T &rlarr;Y and X &rlarr; F between the torsion class T of R-modules and the torsion-free class Y of S-modules and between the torsion class X of S-modules and the torsion-free class F of R-modules. Denote by (R, V, S) and (S, U, R) the quasi-tilting triples representing these equivalences. We say that (R, V, S) and (S, U, R) are complementary if (T, F) and (X, Y) are torsion theories in R-Mod and S-Mod, respectively. We find necessary and sufficient conditions on the bimodules [SUBR]V[SUBS] and [SUBS]U[SUBR] to have the complementarity of (R, V, S) and (S, U, R). [ABSTRACT FROM AUTHOR]

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair (T , F) of full replete subcategories in a category C , the corresponding full subcategory Z = T ∩ F of trivial objects in C. The morphisms which factor through Z are called Z -trivial, and these form an ideal of morphisms, with respect to which one can define Z -prekernels, Z -precokernels, and short 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 Z is reduced to the 0-object of 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. [ABSTRACT FROM AUTHOR]