### $A$-projective resolutions and an Azumaya theorem for a class of mixed abelian groups

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Glaz and Wickless introduced the class $G$ of mixed abelian groups $A$ which have finite torsion-free rank and satisfy the following three properties: i) ${A}_{p}$ is finite for all primes $p$, ii) $A$ is isomorphic to a pure subgroup of ${\Pi}_{p}{A}_{p}$, and iii) $\mathrm{H}om(A,tA)$ is torsion. A ring $R$ is a left Kasch ring if every proper right ideal of $R$ has a non-zero left annihilator. We characterize the elements $A$ of $G$ such that $E\left(A\right)/tE\left(A\right)$ is a left Kasch ring, and discuss related results.

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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