On the Cotorsion Pair (𝒫1, 𝒟)

Abstract

We consider the perfect cotorsion pair (𝒫1, 𝒟) over a commutative ring (often over a domain) R consisting of modules of projective dimension ≤1 and of divisible modules, respectively. The kernel modules in this cotorsion pair are well kown (Theorem 2.1), and those domains are identified whose injectives are kernel modules (Theorem 2.3). The kernel modules always generate the cotorsion pair (𝒫1, 𝒟); they also cogenerate it if the global dimension of the domain R is finite (Theorems 3.1, 3.2). An analogue of the well-known Faith–Walker theorem on injective modules is proved: an integral domain R is noetherian of Krull dimension 1 if there is a cardinal number λ such that every weak-injective (or divisible) R-module is a direct sum of modules of cardinalities ≤ λ (Theorem 4.4). The proof relies on our Theorem 4.1 which generalizes the Faith–Walker theorem as well as a theorem by Guil and Herzog in [14]. Theorem 4.2 provides a general sufficient criterion for a complete cotorsion pair to be Σ-cotorsion; here Σ-cotorsion means that both classes in the cotorsion pair are closed under direct sums.