The concept of torsion had its origin in the theory of abelian groups. A group is torsion if and only if all its elements have finite order. This notion can be directly extended to modules over an integral domain. In the theory of rings and modules of quotients, there is, associated with each ring of quotients of a ring R, a notion of torsion for modules over R ([Stenström, Rings and modules of quotients, LNM 237, Springer, 1971], II.3).

Several authors, as Dlab in [The concept of a torsion module, Amer. Math. Monthly 75 (1968) 973-976] and Levy in [Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15 (1963) 132-151], extended this notion to modules over more general rings. However these definitions are too closely linked with some particular classes of modules and, apparently, not connected.

In 1966 Dickson [A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966) 223-235] introduced a notion of torsion theory in abelian categories which generalizes and unifies the existing ones. As Herrlich and Strecker refer in [Category Theory, Allyn and Bacon, Boston, 1973], quoting Bass, "Virtually all algebraic notions in Category Theory are parodies of their parents in the most classical of categories... the category of left A-modules."

The extension of fundamental ideas and constructions of one area of Mathematics to another one is one of the aims of the Theory of Categories. This was Barr's purpose in [Non-abelian torsion theories, Canad. J. Math. 25 (1973) 1224-1237], where he presented a definition of torsion theory for non-abelian categories and a great variety of examples encompassed by these theories. Also Cassidy, Hébert and Kelly, in 1985, following a different path, introduced another definition of torsion theory which arose from a close analysis of the relations between reflective subcategories and factorization systems of a category.

This dissertation, which is mainly based on [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329], summarizes some of the results in this area.

In the first chapter, where we use essentially [Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966) 223-235] and [Dickson, On torsion classes of abelian groups, J. Math. Soc. Japan 17 (1965) 30-35], we present an approach to torsion theories in abelian categories. In the first section we present Dickson's definition and stress the analogy between the theory of torsion/torsion free subcategories in this context and the one of connectednesses/disconectednesses in the category of topological spaces as it appears in [Arhangel'skii and Wiegandt, Connectednesses and disconnectednesses in Topology, Gen. Top. Appl. 5 (1975) 9-33]. The concept of hereditary torsion theories is a very important one. Indeed, they are equivalent to, for example, "idempotent filters of ideals" in [Bourbaki, Éléments de Mathématique, fasc. 27, Chap.1, 2 (1961)] and [Gabriel, Des catégories abéliennes, Bull. Soc. Mat. France 90 (1962) 323-448], nowadays called Gabriel topologies, "left exact torsion radicals" in [Maranda, Injective Structures, Trans. Amer. Math. Soc. 110 (1964)], "idempotent kernel functors" in [Goldman, Rings and modules of quotients, J. Algebra 13 (1969)] and to "left exact reflectors" in [Gabriel, Des catégories abéliennes, Bull. Soc. Mat. France 90 (1962) 323-448]. This is the topic of the second section where we present a characterization theorem extracted from [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] and [B. Stenström, Rings and modules of quotients, LNM 237, Springer, 1971]. Finally in Section 3, where we use mainly [Dickson, On torsion classes of abelian groups, J. Math. Soc. Japan 17 (1965) 30-35], we present, with some detail, a classification in the category of abelian groups of all torsion subcategories contained in the subcategory of torsion groups (in the classical sense) and of all hereditary torsion subcategories.

The second chapter starts with the definitions of prefactorization and factorization systems as well as the statements and proofs of the results that will enable us to establish the one-to-one correspondence between reflective subcategories and prefactorization systems of a category. All these results are from [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329], being 1.8 a generalization of 2.4 in [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329]. We point out that this bijection develops an idea referred by Bousfield in ([Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1977) 207-220], 2.5). Observation 2.5 of [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] led us to the study of the links between the notions introduced in the preceding section and the categories of fractions. Under very general conditions on a category, namely the finite well-completeness, it is proved in [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] that the bijection referred to above is a bijection between reflective subcategories and factorization systems. This is the subject we deal with in 3.2, 3.3 and 3.4. In 3.5 we present an easy proof given by Cassidy, Hébert and Kelly in [Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] of a result of Day [On adjoint-functor factorization, LNM 420, Springer, 1974]. Section 4 as well as the first one consist of a short account of the essential steps in [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] towards the formulation of the notion of torsion theory for categories with initial and terminal objects. There we present a detailed description of the reflective and the coreflective factorization systems associated with a factorization system and exhibit some examples. At this point it is possible to study torsion theories in categories with initial and terminal objects. This is the topic of Section 5. The Definition 5.2, which can be formulated in any category, was suggested by the study of [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] and aims to stress the differences and the similarities with the characterization of torsion in abelian categories. In 5.4 we prove that this is a generalization of the definition of Cassidy, Hébert and Kelly. The remaining results are from [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] with exception of 5.7 and of example (c) where we study the torsion theories in the category of graphs and in preorder categories defined by complete lattices, respectively. Section 6 contains a brief survey of torsion theories in categories without zero object as well as sufficient conditions for a category without zero object to have only as torsion theories the trivial ones. As a direct consequence we conclude that the notion of torsion becomes trivial in topological categories giving us a justification for the fact of being the analogue theory of connectednesses/disconnectednesses that plays an important role in such categories. Except for 7.10 (ii), which was inspired by the similar result for abelian categories, all the results in section 7 are from [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] and deal with torsion theories in categories with a zero object. In 8.5, 8.6, 8.7, 8.8 and 8.10 we collect results of an approach given by Tholen in [Factorizations, fibres and connectedness, Proc. Int. Conf. Categorical Topology, (Toledo, 1983), Heldermann, 549-566] for the study of connections in the context of factorizations of sources and apply them to the special case we are interested in: the torsion theories in categories with zero object. With these results, and still inspired in [Tholen, Factorizations, fibres and connectedness, Proc. Int. Conf. Categorical Topology, (Toledo, 1983), Heldermann, 549-566], we obtain Theorem 8.11 and its corollaries 8.12 and 8.13, which give us a characterization of torsion theories in terms of E-connections and E-reflections. Finally, we are able to relate in proposition 8.14 the definition given by Cassidy, Hébert and Kelly with Barr's definition.

The third chapter deals with torsion theories and localizations. In Section 1 we first recall the definition of certain types of reflectors given in [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] and its characterizations, giving particular attention to the reflectors with stable units (1.6) whose characterization is based on results of [Korostenski and Tholen, Prelocalizations, preprint] and on the observation that condition 1.9 (iv) in this paper coincides with the definition of reflector with stable units. The Example 1.8 is a detailed proof of Theorem 1 of [Cagliari and Mantovani, Localizations in universal topological categories, Proc. Amer. Math. Soc. 103 (1988) 639-640]. The relation between localizing subcategories and hereditary torsion theories is studied in Section 2. Its main goal is to present a result of [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329] where it is shown that, in abelian categories with enough injectives, there exists a one-to-one correspondence between localizing subcategories and hereditary torsion theories. It also contains some examples of non-abelian categories where this correspondence is not a bijection. The third and last section is concerned with the idea of setting up a link with the classical results on localizations suggested in the introduction of [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (1985) 287-329]. For that, our main tools are two results of [Borceux and Kelly, On locales of localizations, J. Pure Appl. Algebra 46 (1987) 1-34] which we recall in 3.2 and 3.4. With these results we describe localizing subcategories of some special categories in terms of an adequate topology for each case. We also make a short reference to the case of toposes.