FC-PURE IINJECTIVITY AND FC-PURE FLATNESS

In this paper, we consider relation between FC-purity and various purities. In particular, it is shown that every left R-module is FC-pure injective (projective) if and only if R is a left K¨othe ring. Moreover, if all idempotents of R are central, then every left and right R-module is FC-pure injective (projective) if and only if R is an Artinian principal ideal ring. Also, the notions of FC-pure and I-pure flat modules are introduced and several characterizations are given. We prove that a ring R with central idempotents is Artinian if and only if every I-pure flat R-module is direct sum of finitely presented cyclic modules