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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.

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

We consider when a single submodule and also when every submodule of a module M over a general ring R has a unique closure with respect to a hereditary torsion theory on Mod-R.

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.

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.

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.

We show that, if E is a Landweber exact ring spectrum, then the category of E_*E-comodules is equivalent to the localization of the category of BP_*BP-comodules with respect to the hereditary torsion theory of v_n-torsion comodules, where n is the height of E. In particular, the category of E(n)_*E(n)-comodules is equivalent to the category of (v_n^{-1}BP)_*(v_n^{-1}BP)-comodules. We also prove structure theorems for E_*E-comodules; we show every E_*E-comodule has a primitive, we classify the invariant radical ideals, and we prove a version of the Landweber filtration theorem., Comment: 24 pages

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.

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.

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.

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

We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras., Comment: 11 pages, revised version (ver. 2)

We establish relations between two extensions of Dickson's torsion theory, one introduced by Barr and the other by Cassidy, Hubert and Kelly. These two notions are studied in a general category and we show that categories with zero object are, in some sense, the ones where the second notion is relevant. In these categories, under suitable completeness conditions, we characterize torsion theories in terms of reflections and connections. A characterization of the torsion-free subcategories which generalizes the one of Dickson is also presented.

This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, \v{C}ech and local cohomology modules over commutative rings which are not necessarily Noetherian. We also compare these approaches of grade., Comment: to appear in Communication in Algebra

We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of Komarnitskiy. We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings. Визначено кільця, для яких кожна теорія скруту з успадкуванням є теорією S-скруту у сенсі Комарницького. Показано, що такі кільця допускають первинний розклад. Комарницький отримав цей результат у частинному випадку лівих дуо-кілець.

We introduce a relativized notion of (semi)normalcy for categories that come equipped with a proper stable factorization system, and we use radicals (in the sense of module theory) and normal closure operators in order to study torsion theories in such categories. Our results generalize and complement recent studies in the realm of semi-abelian and, in part, homological categories. In particular, we characterize both, torsion-free and torsion classes, in terms of their closure under extensions. We pay particular attention to the homological and, for our purposes more importantly, normal categories of topological algebra, such as the category of topological groups. But our applications go far beyond the realm of these types of categories, as they include, for example, the normal, but non-homological category of pointed topological spaces, which is in fact a rich supplier for radicals of topological groups. http://www.sciencedirect.com/science/article/B6WH2-4HK04RW-1/1/18b9f9a869360fcbae5bb6200eabdf20

In this survey article, we indicate how some of the recent ideas and techniques introduced in the study of infinite rank Butler groups can be successfully used in the investigation of the homological dimensions of torsion-free modules over integral domains and, in particular, over valuation domains.

In [K] Y.Kurata proved that the Goldie torsion theory splits centrally for dual rings. Here we extend his result to semilocal rings with left essential socle such that Soc (RR) ⊆ Soc (RR). An example will demonstrate that our observation extends Kurata’s result. All rings are associative ring with unit and all R-modules are unital. The singular submodule of a left R-module M is denoted by Z(RM). We abbreviate S := Soc (RR) and J := Jac (R) for the left socle resp. the Jacobson radical. We denote the left Goldie torsion theory, that is the hereditary torsion theory generated by all singular left R-modules, by τG and the torsion submodule of a module M by τG(M). τG is said to be centrally splitting if τG(R) is a ring direct summand of R. A classical result of Allin and Dickson [AD] states that τG is centrally splitting for a ring R if and only if R is a direct product of a semisimple ring and a ring with essential left singular ideal. Lemma 1. Let R be a ring with essential left socle S. Then (1) S = S ⊕ (S ∩ Z(RR)), where S is projective and R/S is τG-torsion. (2) J is τG-torsion if and only if S J = 0. Proof. The socle can be decomposed as S = S0 ⊕ S1 where S1 := S ∩ Z(RR) and S0 is a projective left R-module. S ⊆ S0, because for x, y ∈ S with x = x0+x1 and y = y0+y1 where x0, y0 ∈ S0 and x1, y1 ∈ S1. The product xy = xy0 ∈ S0 as xy1 ∈ SZ(RR) = 0. Thus S ⊆ S0 holds and there exists a left module S such that S0 = S ⊕ S. We have SS ⊆ S∩ S = 0. If RS is essential in RR, then S becomes singular (as it is annihilated by S) and must be zero as it is also projective. Thus S0 = S . Also R/S becomes τG-torsion as S1 ' S/S and R/S are 1991 Mathematics Subject Classification. 16S90.

We characterize reflexive modules over $QF- 3^{\prime}$ rings using a linear compactness condition relative to the Lambek torsion theory, and we also give a necessary and sufficient condition for a left $QF- 3^{\prime}$ maximal quotient ring to be right $QF- 3^{\prime}$.

The equality or non-equality of radical classes, torsion classes and connectednesses is investigated in a category that could be any of the concrete categories in which a radical theory, connectednesses theory or a torsion theory have been developed. Among others, it is shown that every connectedness is a radical class. The converse is not true. If, however, a radical class is strict, then it is a connectedness. Furthermore it is proved that every torsion class is a radical class and the converse is not necessarily true. Every radical class with the ADS-property is a torsion class.

This paper continues the study of codivisible modules, whose definition is a “dualization” of Lambek's concept [4] of a divisible module relative to a torsion theory. The main purpose of this work is to give a solution to the following problem posed by Bland [2]: “It would be interesting to know under what conditions the universal existence of codivisible covers implies that of projective covers.” The if-and-only-if nature of the solution, which is given in Theorems 2 and 4, shows that our sufficient conditions are “best possible” conditions. The method of the solution introduces the concept of a pseudo-hereditary torsion theory, which may be of interest in its own right; in particular, every hereditary torsion theory and every faithful torsion theory is pseudo-hereditary. The main results for a pseudo-hereditary torsion theory (T, F) relate the conditions, R/T(R) is semiperfect and R/T(R) is left perfect, to the existence of codivisible covers (see Theorems 1 and 3 and Corollaries 1 and 2).

Simmons [1986] has defined a new filtration for torsion theories over module categories. In this note we show that this filtration coincides with the Gabriel filtration for a large class of rings.

In this paper, the following condition for a torsion theory in the sense of S. E. Dickson is examined: ( P \text {P} ) Every nonzero torsionfree module contains a nonzero projective submodule. A special relationship between condition ( P \text {P} ) and Goldie’s torsion theory is shown; and the rings, for which every nontrivial torsion theory satisfies condition ( P \text {P} ), are classified.

We introduce and study the notion of torsion theory in the non-abelian context of homological categories, and we investigate the properties of the corresponding closure operator. We then consider several new examples of torsion theories in the category of topological groups and, more generally, in any category of topological semi-abelian algebras. We finally characterize the hereditary torsion theories, and we analyse a new example in the homological category of crossed modules. (c) 2006 Elsevier Inc. All rights reserved.

When we consider modules A over a ring R which is not a commutative integral domain, the usual torsion theory becomes somewhat inadequate, since zero-divisors of R are disregarded and since the torsion elements of A do not in general form a submodule. In this paper we shall try to remedy such defects by modifying the fundamental notions such as torsion modules, divisible modules, etc.