Given a flat injective ring epimorphism u : R → U between commutative rings, we consider the Gabriel topology G associated to u and the class DG of G-divisible modules. We prove that DG is an enveloping class if and only if it is the tilting class corresponding to the 1-tilting module U ⊕ U/R and for every ideal J ∈ G the quotient rings R/J are perfect rings. Equivalently, p. dim U ≤ 1 and the discrete quotient rings R/RJ of the topological ring R = End(U/R) are perfect rings.Moreover, we show that every enveloping 1-tilting class over a commutative ring arises from a flat injective ring epimorphism.
We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (A, T ) provides for covers, that is when the class A is a covering class. We use Hrbek's bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if G is the Gabriel topology associated to the 1-tilting cotorsion pair (A, T ), and RG is the ring of quotients with respect to G, we show that if A is covering then G is a perfect localisation (in Stenström's sense [Ste75]) and the localisation RG has projective dimension at most one. Moreover, we show that A is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈ G. Rings satisfying the latter two conditions are called G-almost perfect.
We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( A , T ) (\mathcal{A},\mathcal{T}) provides for covers, that is when the class 𝒜 is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring 𝑅 and the faithful finitely generated Gabriel topologies on 𝑅. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 is the Gabriel topology associated to the 1-tilting cotorsion pair ( A , T ) (\mathcal{A},\mathcal{T}) , and R G R_{\mathcal{G}} is the ring of quotients with respect to 𝒢, we show that if 𝒜 is covering, then 𝒢 is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R G R_{\mathcal{G}} has projective dimension at most one as an 𝑅-module. Moreover, we show that 𝒜 is covering if and only if both the localisation R G R_{\mathcal{G}} and the quotient rings R / J R/J are perfect rings for every J ∈ G J\in\mathcal{G} . Rings satisfying the latter two conditions are called 𝒢-almost perfect.
In this paper we consider the class P1(R) of modules of projective dimension at most one over a commutative ring R and we investigate when P1(R) is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether P1(R) is covering necessarily implies that P1(R) is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring R. This gives an example of a cotorsion pair (P1(R), P1(R) ⊥ ) which is not necessarily of finite type such that P1(R) satisfies Enochs' Conjecture. Moreover, we describe the class lim − → P1(R) over (not-necessarily commutative) rings which admit a classical ring of quotients.
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