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Ideal ring extensions and trusses
Ryszard R. Andruszkiewicz,
Tomasz Brzezinski 
,
Bernard Rybołowicz
,
Bernard Rybołowicz
Journal of Algebra, Volume: 600, Pages: 237 - 278
Swansea University Author:
Tomasz Brzezinski
-
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DOI (Published version): 10.1016/j.jalgebra.2022.01.038
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...
| Published in: | Journal of Algebra |
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| ISSN: | 0021-8693 |
| Published: |
Elsevier BV
2022
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| Online Access: |
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| 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. |
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| Keywords: |
Truss; Ring; Extension |
| College: |
Faculty of Science and Engineering |
| 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 |