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 Enochscotorsion. (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 ≤ 1 are of projective dimension ≤ 1. (vii) The cotorsion pairs (P 1 , D) and (F 1 , 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.

Matlis category equivalences for a ring epimorphism

“…According to [11,Proposition 3.11], any 1-cotilting torsion pair of cofinite type in the category of modules over a commutative ring is hereditary. By Lemma 5.1 (4) =⇒ (6), it follows that fd R U = 0.…”

“…Let us mention two further generalizations of the Matlis category equivalences in two different directions, which appeared in the two recent papers [20,6]. In the paper [20,Section 5], the two Matlis additive category equivalences were constructed for a localization S −1 R of a commutative ring R with respect to a multiplicative subset S ⊂ R. Injectivity of the map R −→ S −1 R was not assumed, but the assumption that the projective dimension of the R-module S −1 R does not exceed 1 was made.…”

“…In the paper [20,Section 5], the two Matlis additive category equivalences were constructed for a localization S −1 R of a commutative ring R with respect to a multiplicative subset S ⊂ R. Injectivity of the map R −→ S −1 R was not assumed, but the assumption that the projective dimension of the R-module S −1 R does not exceed 1 was made. In the paper [6,Section 4], the first Matlis category equivalence was constructed for certain injective epimorphisms of noncommutative rings R −→ Q, where Q is the localization of R with respect to a one-sided Ore subset of regular elements.…”

“…=⇒ (5) are obvious. (5) =⇒(6) Let A be a left R-module. From the exact sequence(8) we see that the left R-module Tor R 1 (U, A) is a submodule of the left R-module TorR 1 (K • , A).…”

On a partially ordered set associated to ring morphisms

“…One of the mappings appearing in this factorization of ϕ is a ring epimorphism ϕ| T : R → T , which still corresponds to the pair (a, M ). Ring epimorphisms, that is, epimorphisms in the category Ring, currently play a predominant role in Homological Algebra [1,6,14,15,16], in particular left flat morphism, that is, when the codomain is a flat left R-module. The functor Hom(−) is not representable (Section 2).…”