Abstract
For an Abelian group \(A\), viewed as a module over its endomorphism ring \(E(A)\), the near-ring \(\mathcal{M}_{E(A)}(A)\) of homogeneous mappings is defined as the set of mappings \(\{f\colon A\to A \mid f(\varphi a)=\varphi f(a)\) for all \(\varphi\in E(A)\) and \(a\in A\}\) with the operations of addition and composition (as multiplication). It is proved that the problem of describing some classes of mixed Abelian groups with the property \(\mathcal{M}_{E(A)}(A)=E(A)\) reduces to the cause of torsion-free Abelian groups. Abelian groups with this property are found in the class of strongly indecomposable torsion-free Abelian groups of finite rank and torsion-free Abelian groups of finite rank coinciding with their pseudosocle.
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Notes
See Sec. 4.1 below.
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 872-883 https://doi.org/10.4213/mzm12942.
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Lyubimtsev, O.V. Endomorphism of Abelian Groups as Modules over Their Endomorphism Rings. Math Notes 109, 909–917 (2021). https://doi.org/10.1134/S0001434621050242
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DOI: https://doi.org/10.1134/S0001434621050242