Abstract
Let [Formula: see text] be a commutative ring and [Formula: see text] a given multiplicative closed subset of [Formula: see text]. In this paper, we introduce the new concept of [Formula: see text]-torsion exact sequences (respectively, [Formula: see text]-torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules.
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