Pretorsion theories in general categories

Abstract

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.