### Strong $\mathcal{S}$-groups

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An exact sequence $0\to A\to B\to C\to 0$ of torsion-free abelian groups is quasi-balanced if the induced sequence $$0\to \mathbf{Q}\otimes Hom(X,A)\to \mathbf{Q}\otimes Hom(X,B)\to \mathbf{Q}\otimes Hom(X,C)\to 0$$ is exact for all rank-1 torsion-free abelian groups $X$. This paper sets forth the basic theory of quasi-balanced sequences, with particular attention given to the case in which $C$ is a Butler group. The special case where $B$ is almost completely decomposable gives rise to a descending chain of classes of Butler groups. This chain is a generalization of the chain of Kravchenko classes that arise from balanced...

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...

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket $R$-module is $R$ tensor a bracket group.

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