Geodesic
$\tau$-complemented and $\tau$-supplemented modules
Algebra and discrete mathematics, no. 3 (2006), pp. 1-16

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Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homological 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, Generalov, 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.
Keywords: proper classes; neat and coneat submodules; $\tau$-supplemented, $\tau$-complemented and $\tau$-semiperfect modules. 
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Khaled Al-Takhman; Christian Lomp; Robert Wisbauer. $\tau$-complemented and $\tau$-supplemented modules. Algebra and discrete mathematics, no. 3 (2006), pp. 1-16. http://geodesic.mathdoc.fr/item/ADM_2006_3_a0/