SCALAR MULTIPLICATION OF PROPER CLASSES OF SHORT EXACT SEQUENCES

Throughout this paper we assume that R is an associative ring with an identity element. It is well known that Pext(C,A) = ⋂ n6=0 nExt(C,A) for every abelian groups C,A (see e.g. [7]). In other words for the class S of all pure short exact sequences and Abs of all short exact sequences we have S = ⋂ n6=0 nAbs. The short exact sequences of nAbs are described in [7]. Two generalizations of purity for modules are considered in [20] which coincide for modules over Prfer rings. In this paper we prove that rA is a proper class for every r ∈ R and for every proper class A of short exact sequences of R-modules for a principal ideal domain R if  = A where  = {E : rE ∈ A for some 0 6= r ∈ R}. As a corollary of this fact we obtain the well-known fact that all neat short exact sequences form a proper class. Pure-projective and pure-injective modules are described in [20]. We study nA-injective objects and nA-projective objects in the case of abelian groups.