Rings over which every RD-projective module is a direct sums of cyclically presented modules

Abstract We study direct-sum decompositions of RD -projective modules. In particular, we investigate the rings over which every RD -projective right module is a direct sum of cyclically presented right modules, or a direct sum of finitely presented cyclic right modules, or a direct sum of right modules with local endomorphism rings (SSP rings). SSP rings are necessarily semiperfect. For instance, the superlocal rings introduced by Puninski, Prest and Rothmaler in [28] and the semilocal strongly π -regular rings introduced by Kaplansky in [24] are SSP rings. In the case of a Noetherian ring R (with further additional hypotheses), an RD -projective R -module M turns out to be either a direct sum of finitely presented cyclic modules or of the form M = T ( M ) ⊕ P , where T ( M ) is the torsion part of M (elements of M annihilated by a regular element of R ) and P is a projective module.