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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

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 $\mathfrak{Qcoh}(X)$.

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.

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.

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.

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.

Differential torsion theories are introduced and it is shown that for a hereditary torsion theory τ every derivation on an R-module M has a unique extension to its module of quotients if and only if τ is a differential torsion theory. Dually, we show that when τ is cohereditary, every derivation on M can be lifed uniquely to its module of coquotients.

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for $P^{\centerdot}$ to be a tilting complex. Finally, in the case of artin algebras, we give one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions., 17 pages, AMSLaTeX

Locally finitely presented Grothendieck abelian categories are shown to be just the “elementary” localizations of Module categories. All categories V are assumed to be Grothendieck abelian (e.g. see [l, 21 or [5]). 55’ is locally finifeely presented if 9? has a generating family Ce = {Gi: i E I} with each Gi a finitely presented object of %. I have shown elsewhere [4] that in such categories one may define a canonical first-order (many-sorted) logic which behaves well one may prove Los’ Theorem, produce K-Saturated extensions of structures etc. Given a torsion theory (= a hereditary torsion theory in the terminology of [5]) (T, S) on Q and C in %‘, Q$ will be the filter of T-dense subobjects of C (where C’s C is T-dense if C/C’ is a member of the torsion class T). %? will denote the quotient category of V with respect to (9,s) and ( -)swill denote the T-localization functor from %’ to %? Tr will be the functor taking objects C of % to their T-torsion subobjects. We say that C is T-finitely generated (T-fig.) if C contains a finitely generated T-dense subobject. Recall that a torsion theory (9,9) is of finite type if for every Gi E 59 (as above), %$I has a cofinal subset of finitely generated members equivalently (by an easy argument) iff this condition holds on %s for any finitely generated c E %. We shall say that a torsion theory (9,T) is elementary if: (i) (9, 9) is of finite type; (ii) for each Gi, Gi E ‘3 (as above) and r: Gi --, Gi with im r E %z, ker r is T-finitely generated (that is (finitely generated) members of %z are **T-finitely presented” in some sense). It may be shown that this notion does not depend on the choice of ‘3 (directly or as a consequence of the following result). The term “elementary” is justified by the following theorem