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

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.