Pure-periodic modules and a structure of pure-projective resolutions

We investigate the structure of pure-syzygy modules in a pure-projective resolution of any right R-module over an associative ring R with an identity element. We show that a right R-module M is pure-projective if and only if there exists an integer n greater than or equal to 0 and a pure-exact sequence 0 --> M --> P-n --> ... P-0 --> M --> 0 with pure-projective modules P-n,..., P-0. As a consequence we get the following version of a result in Benson and Goodearl, 2000: at module M is projective if M admits an exact sequence 0 --> M --> F-n --> ... F-0 --> M --> 0 with projective modules F-n,..., F-0.