Search

Let A be a complete and cocomplete abelian category with the additional assumption that any direct sum and product of short exact sequences are exact. We explore the relation between balanced pairs and TTF triples in A. The main results are: (1) every balanced pair in A induces a TTF triple; (2) if A has projective covers and injective envelopes, then every TTF triple in A gives rise to a balanced pair, and hence there is a bijection between the equivalence classes of balanced pairs and TTF triples in A ; (3) a balanced pair in A is quasi admissible if and only if its induced TTF triple is centrally splitting. Our first application of these results provide abundant rings over which every balanced pair is quasi admissible, including local rings, commutative semiperfect rings, and commutative Noetherian rings. Another application is the classification of equivalence classes of cohereditary balanced pairs over arbitrary rings. We also present counterexamples to [15, Open questions 3 and 5]. Finally, we prove that the answers to [15, Open questions 2 and 4] are positive for coherent rings with weak global dimension at most one. [ABSTRACT FROM AUTHOR]

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 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]