New results for the additive groups of Hamiltonian rings

Abstract

This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups.

KEYWORDS:

• Abelian groups
• additive groups of rings
• commutative rings
• H-rings
• ideals
• not necessarily associative rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20K99
• 16D25
• 17D99
• 13A15

Acknowledgments

The author is grateful to the anonymous reviewer for his editorial support. The author express his gratitude also to Divya Elumalai for her assistance in communication after some troubles related to mailboxes.

Additional information

Funding

The author’s home university received financial support from the Ministry of Education and Science under subsidy for maintaining the research potential of the Faculty of Mathematics, University of Bialystok.