Abstract
The property that we have termed generalized Bassian is a natural concept for many areas of algebra, namely the existence of an injective homomorphism \(A\to A/I\) for an object (group, ring, module, algebra, etc.) \(A\) with a normal sub-object \(I\) (normal subgroup, ideal, submodule, etc.) forces that \(I\) is a direct summand of \(A\). It is a common generalization of the question is it possible to embed an object (group, ring, module, algebra, etc.) in a proper homomorphic image of itself, originally raised by Bass [3]. Here we study the generalized concept for Abelian groups and achieve a certain deep although not complete characterization of all Abelian groups satisfying this generalized property.
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Chekhlov, A.R., Danchev, P.V. & Goldsmith, B. On the generalized Bassian property for Abelian groups. Acta Math. Hungar. 168, 186–201 (2022). https://doi.org/10.1007/s10474-022-01262-x
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DOI: https://doi.org/10.1007/s10474-022-01262-x