RING EXTENSIONS AND DIMENSIONS OF MODULES

Abstract

Let S ≥ R be a ring extension satisfying (1) S is right R-projective, and (2) RS is flat and SR is projective. We prove that if MS is an S-module, then the equalities id(MS) = id(MR), pd(MS) = pd(MR) and fd(MS) = fd(MR) hold, where id(−),pd(−) and fd( −) denote the injective, projective and flat dimension of a module, respectively. We also prove that if R is QF, right hereditary or right (left) IF, then so is S, but the converse are false.