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Noncrossed Product Matrix Subrings and Ideals of Graded Rings

Öinert, Johan LU and Lundström, Patrik (2009) In Preprints in Mathematical Sciences 2009(10).
Abstract
We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced by the grading. In the end of the article, we show that, given a finite groupoid G, which has a nonidentity morphism, there is a ring, strongly... (More)
We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced by the grading. In the end of the article, we show that, given a finite groupoid G, which has a nonidentity morphism, there is a ring, strongly graded by G, which is not a crossed product over G. (Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
unpublished
subject
keywords
ideals, matrix rings, Category graded rings, crossed products
in
Preprints in Mathematical Sciences
volume
2009
issue
10
pages
14 pages
publisher
Lund University
external identifiers
  • other:LUTFMA-5112-2009
ISSN
1403-9338
project
Non-commutative Analysis of Dynamics, Fractals and Wavelets
Non-commutative Geometry in Mathematics and Physics
language
English
LU publication?
yes
id
566a78fa-6f6c-4f30-95ac-c4171e935b61 (old id 1445605)
alternative location
http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.0997v1.pdf
http://arxiv.org/abs/0907.0997
date added to LUP
2016-04-01 13:07:20
date last changed
2018-11-21 20:12:30
@article{566a78fa-6f6c-4f30-95ac-c4171e935b61,
  abstract     = {We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced by the grading. In the end of the article, we show that, given a finite groupoid G, which has a nonidentity morphism, there is a ring, strongly graded by G, which is not a crossed product over G.},
  author       = {Öinert, Johan and Lundström, Patrik},
  issn         = {1403-9338},
  language     = {eng},
  number       = {10},
  publisher    = {Lund University},
  series       = {Preprints in Mathematical Sciences},
  title        = {Noncrossed Product Matrix Subrings and Ideals of Graded Rings},
  url          = {http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.0997v1.pdf},
  volume       = {2009},
  year         = {2009},
}