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Keywords:
precobalanced sequence; cobalanced sequence; torsion-free image; pure submodule
precobalanced sequence; cobalanced sequence; torsion-free image; pure submodule
Summary:
The class of pure submodules ($\mathcal P$) and torsion-free images ($\mathcal R$) of finite direct sums of submodules of the quotient field of an integral domain were first investigated by M. C. R. Butler for the ring of integers (1965). In this case ${\mathcal P} = {\mathcal R}$ and short exact sequences of such modules are both prebalanced and precobalanced. This does not hold for integral domains in general. In this paper the notion of precobalanced sequences of modules is further investigated. It is shown that as in the case for abelian groups the exact sequence $ 0 \rightarrow M \rightarrow L \rightarrow T \rightarrow 0 $ with torsion $T$ is precobalanced precisely when it is cobalanced and in this case will split if $M$ is torsion-free of rank $1$. It is demonstrated that containment relationships between $\mathcal P$ and $\mathcal R$ for a domain $R$ are intimately related to the issue of when pure submodules of Butler modules are precobalanced. An analogous statement is made regarding the dual question of when torsion-free images of Butler modules are prebalanced.
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