Equivalence of Some Homological Conditions for Ring Epimorphisms

TL;DR

This paper establishes the equivalence of several homological conditions for rings with a classical ring of quotients, extending known results from commutative to certain non-commutative rings.

Contribution

It proves the equivalence of multiple homological conditions for right and left Ore rings with a specific flat dimension condition, generalizing prior results from commutative rings.

Findings

Conditions for flat modules and cotorsion modules are equivalent.

Homological properties like weak-injectivity and projective dimension are shown to coincide.

The results extend known theorems from commutative to certain non-commutative rings.

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 $\le1$. $(vii)$ The cotorsion pairs $(\mathcal{P_1},\mathcal{D})$ and $(\mathcal{F}_1,\mathcal{WI})$ coincide. $(viii)$ Divisible right $R$-modules are weak-injective. This extends a result by Fuchs and Salce (2017) for modules over a commutative ring $R$.