Homomorphisms between \(A\)-projective Abelian groups and left Kasch-rings. (English) Zbl 0931.20043

The authors’ aim is to study classes of abelian groups for which one can prove results in the spirit of Széle’s Theorem describing when a subgroup of the form \(\bigoplus_l\mathbb{Z}/p^n\mathbb{Z}\) of an abelian group is a direct summand. This continues earlier work of the first author done in the case of torsion-free abelian groups of finite rank. The authors consider the class \(\mathcal G\) of all mixed abelian groups \(A\) which have finite torsion-free rank, \(A_p\) is finite for any prime \(p\), \(A\) is isomorphic to a pure subgroup of \(\prod_pA_p\) and where \(\text{Hom}(A,tA)\) is torsion. They call a ring \(R\) left Kasch ring if every proper right ideal of \(R\) has a non-zero left annihilator. The groups \(A\in{\mathcal G}\) for which \(E(A)/tE(A)\) is a left Kasch ring are investigated (Theorem 2.2). The authors further describe the groups \(A\in{\mathcal G}\) which are equal to the direct sum of a finite group and an \(A\)-projective group of finite \(A\)-rank (Theorem 3.4). Two useful examples complete the general study.
Reviewer: L.Beran (Praha)


20K21 Mixed groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups
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