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Ideal ring extensions and trusses

Ryszard R. Andruszkiewicz, Tomasz Brzezinski Orcid Logo, Bernard Rybołowicz

Journal of Algebra, Volume: 600, Pages: 237 - 278

Swansea University Author: Tomasz Brzezinski Orcid Logo

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Abstract

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent...

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Published in: Journal of Algebra
ISSN: 0021-8693
Published: Elsevier BV 2022
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa60087
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Abstract: It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss T and an element of this truss one can associate a ring and its extension by integers in which T is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei (L. Redei (1952) [24]) or by self-permutable bimultiplications of Mac Lane (S. Mac Lane (1958) [18]), that is, as integral homothetic extensions. The correspondence between homothetic ring extensions and trusses is used to classify fully up to iIt is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss T and an element of this truss one can associate a ring and its extension by integers in which T is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei (L. Redei (1952) [24]) or by self-permutable bimultiplications of Mac Lane (S. Mac Lane (1958) [18]), that is, as integral homothetic extensions. The correspondence between homothetic ring extensions and trusses is used to classify fully up to isomorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.somorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.
Keywords: Truss; Ring; Extension
College: College of Science
Funders: The research of the first two authors is partially supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01115. The third author is supported by the EPSRC research grant EP/V008129/1 and was supported by the Swansea University Research Excellence Scholarship (SURES).
Start Page: 237
End Page: 278