2009/04/29 by Johan Öinert, Öinert, Johan
Mathematics · #13A02 #16S35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:13A02 #msc:16S35
paper · pdf · doi:10.48550/arxiv.0904.4661
arxiv created 2009/04/29 · openalex publication_date 2009/04/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring R = \bigoplusg∈ G Rg the grading group G acts, in a natural way, as automorphisms of the commutant of the neutral component subring Re in R and of the center of Re. We show that if R is a strongly G-graded ring where Re is maximal commutative in R, then R is a simple ring if and only if Re is G-simple (i.e. there are no nontrivial G-invariant ideals). We also show that if Re is commutative (not necessarily maximal commutative) and the commutant of Re is G-simple, then R is a simple ring. These results apply to G-crossed products in particular. A skew group ring Re \rtimesσ G, where Re is commutative, is shown to be a simple ring if and only if Re is G-simple and maximal commutative in Re \rtimesσ G. As an interesting example we consider the skew group algebra C(X) \rtimes_h ℤ associated to a topological dynamical system (X,h). We obtain necessary and sufficient conditions for simplicity of C(X) \rtimes_h ℤ with respect to the dynamics of the dynamical system (X,h), but also with respect to algebraic properties of C(X) \rtimes_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.