Noncrossed Product Matrix Subrings and Ideals of Graded Rings

Summary, in English

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: Johan Öinert; Patrik Lundström
• Department/s: Mathematics (Faculty of Engineering)
• Publishing year: 2009
• Language: English
• Publication/Series: Preprints in Mathematical Sciences
• Volume: 2009
• Issue: 10
• Links: http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.0997v1.pdf; http://arxiv.org/abs/0907.0997
• Document type: Journal article
• Publisher: Lund University
• Topic: Mathematics
• Keywords: ideals; matrix rings; Category graded rings; crossed products
• Status: Unpublished
• Project: Non-commutative Analysis of Dynamics, Fractals and Wavelets; Non-commutative Geometry in Mathematics and Physics
• Research group: Non-commutative Geometry
• ISBN/ISSN/Other: ISSN: 1403-9338; LUTFMA-5112-2009