Baer cotorsion Pairs

Abstract

LetR be a unital associative ring and $$\mathfrak{V},\mathfrak{W}$$ two classes of leftR-modules. In [St3] the notion of a ( $$\mathfrak{V},\mathfrak{W}$$ ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses $$\mathcal{V} \subseteq \mathfrak{V} and \mathcal{W} \subseteq \mathfrak{W}$$ is called a ( $$\mathfrak{V},\mathfrak{W}$$ ) pair if it is maximal with respect to the classes $$\mathfrak{V},\mathfrak{W}$$ and the condition Ext R 1 (V, W)=0 for all $$V \in \mathcal{V} and W \in \mathcal{W}$$ . In this paper we study $$\mathfrak{T}\mathfrak{f},\mathfrak{T}$$ pairs whereR = ℤ and $$\mathfrak{T}\mathfrak{f}$$ is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every $$\mathfrak{T}\mathfrak{f},\mathfrak{T}$$ pair is singly cognerated underV=L.