ON φ-EXACT SEQUENCES AND φ-PROJECTIVE MODULES

Let R be a commutative ring with prime nilradical Nil(R) and M an R-module. Define the map phi : R -> R-Nil(R) by (r) = r/1 for r is an element of R and psi : M -> M-Nil(R) by psi(x) = x/1 for x is an element of M. Then psi(M) is a phi(R)-module. An R-module P is said to be phi-projective if psi(P) is projective as a phi(R)-module. In this paper, phi-exact sequences and-projective R-modules are introduced and the rings over which all R-modules are phi-projective are investigated.