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mixed Abelian group; endomorphism ring; Kasch ring; $A$-solvable group
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) $\mathop {\mathrm Hom}\nolimits (A,tA)$ 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 $E(A)/tE(A)$ is a left Kasch ring, and discuss related results.
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