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E. Enochs began the study of injective covers in [3],characterizing when any lefti?-module has an injectivecover. This happens ifand onlyifR is aleft noetherian ring. Torsion injective covers and torsionfreeinjective covers were introduced by Ahsan and Enochs in [2]and [1]respectively,in the context of the Goldie torsion theory. B. Torrecillasin [13] defined r-torsionfree r-injective covers and r-injectivecovers for x a hereditary torsion theory. These covers have been studied in [6] and [7]. In thispaper, r-torsion r-injectivecovers and envelopes are studied for x any torsion theory. Then we construct relativehomological algebra by means of complexes with thiskind of covers and envelopes. In Section 3, we find necessary and sufficientconditions for the existence of r-torsion r-injective covers and envelopes forany module. In the aim of thedescomposition theorem of abelian groups in divisibleand reduced part, we give a torsion theoretical version in terms of r-torsion r-injectivemodules as divisiblesones, Proposition 2, relating such descomposition with certain condition on the existence of r-torsion r-injectivecovers. The existence of r-torsion r-injectiveenvelopes is given in Theorem 2. In Section 4, we study the balance (see [5]) of the functor Hom(―,―) relativeto theclass of r-torsionr-injectivemodules. When thebalance is given, itispossible tointroduce leftderived functor of Hom(―, ―)by using resolvents and resolutions of r-torsion r-injectivemodules. Left and right relativeglobal dimension of the ring R are defined and analysed.

Torsion injective covers and resolvents