Simple semigroup graded rings
(2015) In Journal of Algebra and Its Applications 14(7).- Abstract
- We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves'. We also point out how Jespers' result... (More)
- We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/5385887
- author
- Nystedt, Patrik and Öinert, Johan LU
- organization
- publishing date
- 2015
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Semigroup graded ring, partial skew group ring, simplicity
- in
- Journal of Algebra and Its Applications
- volume
- 14
- issue
- 7
- article number
- 1550102
- publisher
- World Scientific Publishing
- external identifiers
-
- wos:000353552200006
- scopus:84928555107
- ISSN
- 0219-4988
- DOI
- 10.1142/S0219498815501029
- language
- English
- LU publication?
- yes
- id
- 30488f71-5a66-4169-b32e-ee753a9d8872 (old id 5385887)
- date added to LUP
- 2016-04-01 10:52:16
- date last changed
- 2022-03-12 17:50:47
@article{30488f71-5a66-4169-b32e-ee753a9d8872,
abstract = {{We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.}},
author = {{Nystedt, Patrik and Öinert, Johan}},
issn = {{0219-4988}},
keywords = {{Semigroup graded ring; partial skew group ring; simplicity}},
language = {{eng}},
number = {{7}},
publisher = {{World Scientific Publishing}},
series = {{Journal of Algebra and Its Applications}},
title = {{Simple semigroup graded rings}},
url = {{http://dx.doi.org/10.1142/S0219498815501029}},
doi = {{10.1142/S0219498815501029}},
volume = {{14}},
year = {{2015}},
}