Equivalence of Some Homological Conditions for Ring Epimorphisms

Abstract

Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q=R[S^{-1}]=[S^{-1}]R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q)=0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $(ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii)$ $h$-divisible right $R$-modules are weak-injective. $(iv)$ Homomorphic images of weak-injective right $R$-modules are weak-injective. $(v)$ Homomorphic images of injective right $R$-modules are weak-injective. $(vi)$ Right $R$-modules of weak dimension $\le 1$ are of projective dimension $\le 1$. $(vii)$ The cotorsion pairs $({\mathcal{P}}_1,\mathcal{D})$ and $({\mathcal{F}}_1,\mathcal{W}\mathcal{I})$ coincide. $(viii)$ Divisible right $R$-modules are weak-injective. This extends a result by Fuchs and Salce (2017) for modules over a commutative ring $R$.

18 pages