PERIODIC SHORT EXACT SEQUENCES AND PERIODIC PURE-EXACT SEQUENCES

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math. 196(1) (2000) 45-67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 -> M -> P -> M -> 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 -> M -> P -> M -> 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.