|
On fully idempotent homomorphisms of abelian groups
A. R. Chekhlov Tomsk State University, Tomsk, Russia
Abstract:
We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups $A$ and $B$ for which the homomorphism group $\operatorname{Hom}(A,B)$ is fully idempotent. We show that if $B$ is a torsion group or a mixed split group and if at least one of the groups $A$ or $B$ is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups $A$ or $B$ is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups $\operatorname{Hom}(A,A)$ comes down to reduced mixed groups $A$ with dense elementary torsion part.
Keywords:
regular homomorphism, fully idempotent homomorphism, homomorphism group, mixed group, self-small group.
Received: 04.09.2018 Revised: 04.09.2018 Accepted: 19.12.2018
Citation:
A. R. Chekhlov, “On fully idempotent homomorphisms of abelian groups”, Sibirsk. Mat. Zh., 60:4 (2019), 932–940; Siberian Math. J., 60:4 (2019), 727–733
Linking options:
https://www.mathnet.ru/eng/smj3126 https://www.mathnet.ru/eng/smj/v60/i4/p932
|
|