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Abelian groups with monomorphisms invariant with respect to epimorphisms. (English. Russian original) Zbl 1441.20036

Russ. Math. 62, No. 12, 74-80 (2018); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2018, No. 12, 86-93 (2018).
Let \(A\) be an abelian group and \(E:=\mathrm{End}(A)\) its ring of endomorphisms. \(G\) is said to have the R-property (respectively, the L-property) if for every injective endomorphism \(\alpha \) and surjective endomorphism \(\beta \) of \(E\), there exists \(\gamma \in E\) such that \(\beta\alpha =\alpha \gamma \) (respectively, \(\alpha \beta =\gamma \alpha \)).
Among other things the following results are proved.
Theorem 1. If a reduced torsionfree group possesses the R-property or the L-property, then the endomorphism ring of the group is abelian (i.e. its idempotents are central).
Proposition 3. For a divisible group \(D\), the following conditions are equivalent:
1) \(D\)possesses the R-property,
2) all injective endomorphisms of \(D\)are automorphisms,
3) the torsion-free part of \(D\)and each its \(p\)-component have finite rank.
Theorem 3. Let \(A=D\oplus G\), where \(D\neq 0\)is the divisible part of \(A\) and \(G\) has no nonzero divisible homomorphic images contained in \(D\).
1) If the torsion-free part of \(D\)and each \(p\)-component have finite rank, then \(A\)possesses the L-property if and only if \(G\)possesses this property.
2) \(A\)possesses the R-property if and only if \(D\)and \(G\)possess this property.
Direct sums of cyclic groups possessing the R-property or L-property are also described.

MSC:

20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K10 Torsion groups, primary groups and generalized primary groups
20K15 Torsion-free groups, finite rank
16S50 Endomorphism rings; matrix rings
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References:

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