τ-complemented and τ-supplemented modules
Algebra and Discrete Mathematics RESEARCH ARTICLE Number
Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homo-logical algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and this observations were extended to modules by Stenström, Gener-alov, and others. In this note we consider complements and supplements with respect to (idempotent) radicals and study the related proper classes of short exact sequences. 1. Proper classes and
... r classes and τ-supplements With the intention of formalising the theory of Ext functors depending on a specific choice of monomorphisms, Buchsbaum introduced in  certain conditions on a class of monomorphisms which are needed to study relative homological algebra. This lead to he notion of proper classes of monomorphisms and short exact sequences. Well-known examples of such classes are the pure exact sequences which can be defined by choosing a class P of (finitley presented) modules and considering those sequences on which Hom(P, −) is exact for each P ∈ P. These techniques are outlined, for example, in Mishina and Skornjakov , Sklyarenko  and . It was observed in abelian group theory that complement submodules induce a proper class of short exact sequences and this motivated the investigation of such questions for modules over arbitrary rings. First results in this direction were obtained, for example, by Stenström  and Generalov [5, 7]. For a more comprehensive presentation of the results and the sources we refer to E. Mermut's PhD thesis .