Work

Two generalizations of Hopfian Abelian groups

Year: 2025

Type: article

Abstract: This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called relatively Hopfian groups and weakly Hopfian groups, respectively, and to establish some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian p-group G such that [Formula: see text] is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if G is a reduced Abelian p-group such that [Formula: see text] is bounded and [Formula: see text] is Hopfian, then G is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian p-group G with [Formula: see text] an infinite elementary group such that G is not Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated.

Source: Communications in Contemporary Mathematics

DOI: 10.1142/s0219199725500877

Authors: Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith, Patrick W. Keef

Institutions: National Research Tomsk State University, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Technological University Dublin, Whitman College

Language: English

Topic: Rings, Modules, and Algebras

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Junta de Andalucía

Awards: G1737173237, G4863683459