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 ¹_R (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.