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Simple semigroup graded rings

Nystedt, Patrik and Öinert, Johan LU (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)
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author
organization
publishing date
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
publisher
World Scientific
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
2015-05-18 13:24:58
date last changed
2017-01-01 03:58:10
@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.},
  articleno    = {1550102},
  author       = {Nystedt, Patrik and Öinert, Johan},
  issn         = {0219-4988},
  keyword      = {Semigroup graded ring,partial skew group ring,simplicity},
  language     = {eng},
  number       = {7},
  publisher    = {World Scientific},
  series       = {Journal of Algebra and Its Applications},
  title        = {Simple semigroup graded rings},
  url          = {http://dx.doi.org/10.1142/S0219498815501029},
  volume       = {14},
  year         = {2015},
}