$${\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}$$

Abstract

Let R be a unital associative ring and \(\mathfrak{V},\mathfrak{W}\) two classes of left R-modules. In this paper we introduce the notion of a \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pair}}{\text{.}}\) In analogy to classical cotorsion pairs as defined by Salce [10], a pair \({\left( {\mathcal{V},\mathcal{W}} \right)}\) of subclasses \(\mathcal{V} \subseteqq \mathfrak{V}\) and \(\mathcal{W} \subseteqq \mathfrak{W}\) is called a \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pair}}\) if it is maximal with respect to the classes \({\mathfrak{V},\mathfrak{W}}\) and the condition \({\text{Ext}}^1_R (V, W) = 0\) for all \(V \in \mathcal{V}\) and \(W \in \mathcal{W}.\) Basic properties of \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}\) are stated and several examples in the category of abelian groups are studied.