Simple Group Graded Rings and Maximal Commutativity
Författare
Summary, in English
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.
Avdelning/ar
Publiceringsår
2009
Språk
Engelska
Publikation/Tidskrift/Serie
Preprints in Mathematical Sciences
Volym
2009
Issue
6
Länkar
Dokumenttyp
Artikel i tidskrift
Förlag
Lund University
Ämne
- Mathematics
Nyckelord
- crossed products
- Ideals
- graded rings
- simple rings
- maximal commutative subrings
- invariant ideals
- Picard groups
- minimal dynamical systems
Status
Unpublished
Projekt
- Non-commutative Analysis of Dynamics, Fractals and Wavelets
- Non-commutative Geometry in Mathematics and Physics
Forskningsgrupp
- Non-commutative Geometry
ISBN/ISSN/Övrigt
- ISSN: 1403-9338
- LUTFMA-5111-2009