Simple Group Graded Rings and Maximal Commutativity

Öinert, Johan (2009). Simple Group Graded Rings and Maximal Commutativity. Preprints in Mathematical Sciences, 2009, (6)
Download:
URL:
| Unpublished | English
Authors:
Öinert, Johan
Department:
Mathematics (Faculty of Engineering)
Non-commutative Geometry-lup-obsolete
Project:
Non-commutative Analysis of Dynamics, Fractals and Wavelets
Non-commutative Geometry in Mathematics and Physics
Research Group:
Non-commutative Geometry-lup-obsolete
Abstract:
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring R the grading group G acts, in a natural way, as automorphisms of the commutant of the neutral component subring R_e in R and of the center of R_e. We show that if R is a strongly G-graded ring where R_e is maximal commutative in R, then R is a simple ring if and only if R_e is G-simple (i.e. there are no nontrivial G-invariant ideals). We also show that if R_e is commutative (not necessarily maximal commutative) and the commutant of R_e is G-simple, then R is a simple ring. These results apply to G-crossed products in particular. As an interesting example we consider the skew group algebra C(X) ⋊˜h Z associated to a topological dynamical system (X, h). We obtain necessary and sufficient conditions for simplicity of C(X) ⋊˜h Z with respect to the dynamics of the dynamical system (X, h), but also with respect to algebraic properties of C(X) ⋊˜h Z. Furthermore, we show that for any strongly G-graded ring R each nonzero ideal of R has a nonzero intersection with the commutant of the center of the neutral component.
Keywords:
crossed products ; Ideals ; graded rings ; simple rings ; maximal commutative subrings ; invariant ideals ; Picard groups ; minimal dynamical systems
ISSN:
1403-9338
LUP-ID:
3b9937ad-852d-40c6-9ebe-298b061e2827 | Link: https://lup.lub.lu.se/record/3b9937ad-852d-40c6-9ebe-298b061e2827 | Statistics

Cite this