τ-complemented and τ-supplemented modules

Khaled Al-Takhman, Christian Lomp, Robert Wisbauer
2006 Algebra and Discrete Mathematics RESEARCH ARTICLE Number   unpublished
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
more » ... 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 [4] 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 [10], Sklyarenko [13] and [16]. 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 [14] and Generalov [5, 7]. For a more comprehensive presentation of the results and the sources we refer to E. Mermut's PhD thesis [9].
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