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On rings which are sums of subrings

Part of: Rings with polynomial identity Radicals and radical properties of rings

Published online by Cambridge University Press:  21 February 2025

Ryszard R. Andruszkiewicz Open the ORCID record for Ryszard R. Andruszkiewicz [Opens in a new window]  and
Marek Kȩpczyk Open the ORCID record for Marek Kȩpczyk [Opens in a new window]
Ryszard R. Andruszkiewicz
Affiliation:
Faculty of Mathematics, University of Bialystok, Ciolkowskiego 1M, 15–245 Bialystok, Poland e-mail: r.andruszkiewicz@uwb.edu.pl
Marek Kȩpczyk*
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bialystok, Poland
*
e-mail: m.kepczyk@pb.edu.pl
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Abstract

There are presented some generalizations and extensions of results for rings which are sums of two or tree subrings. It is provided a new proof of the well-known Kegel’s result stating that a ring being a sum of two nilpotent subrings is itself nilpotent. Moreover, it is proved that if R is a ring of the form $R=A+B$, where A is a subgroup of the additive group of R satisfying $A^d\subseteq B$ for some positive integer d and B is a subring of R such that $B\in S$, where S is N-radical contained in the class of all locally nilpotent rings, then $R\in S$.

Keywords

Left T-nilpotent ringsum of subrings

MSC classification

Primary: 16R10: $T$-ideals, identities, varieties of rings and algebras 16N40: Nil and nilpotent radicals, sets, ideals, rings 16R50: Other kinds of identities (generalized polynomial, rational, involution)
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 9
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The research of Marek Kȩpczyk was supported by Bialystok University of Technology grant WZ/WI-IIT/2/2023.

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