Equivalence of some homological conditions for ring epimorphisms

Abstract

Let R⊆Q be rings such that the inclusion φ:R→Q is an epimorphism in the category of rings and Q is flat both as a right and as a left R-module. Assume that F.dim(QQ)=0, that is, that every left Q-module has projective dimension 0 or ∞. We prove that the following conditions are equivalent: (i) Flat left R-modules are strongly flat. (ii) Matlis-cotorsion left R-modules are Enochs-cotorsion. (iii)h-divisible left R-modules are weak-injective. (iv) Homomorphic images of weak-injective left R-modules are weak-injective. (v) Homomorphic images of injective left R-modules are weak-injective. (vi) Left R-modules of weak dimension ≤1 are of projective dimension ≤1. (vii) The cotorsion pairs (P1,HD) and (F1,WI) coincide. Moreover, if R is a right and left Ore ring, S is its set of regular elements and Q=R[S−1]=[S−1]R is the classical ring of quotients of R, then the seven conditions above are also equivalent to: (viii) Divisible left R-modules are weak-injective. This extends a result by Fuchs and Salce (2018) [10] for modules over a commutative ring R.