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.