Commutativity and ideals in strongly graded rings. (Q844255)

Let \(G\) be a group with neutral element \(e\). A ring \(R\) is said to be \(G\)-graded if \(R=\bigoplus_{g\in G}R_g\) and \(R_gR_h\subseteq R_{gh}\) for all \(g,h\in G\) where \(\{R_g\}_{g\in G}\) is a family of additive subgroups in \(R\). A \(G\)-graded ring \(R\) is said to be strongly graded by \(G\) if \(R_gR_h=R_{gh}\) for all \(g,h\in G\). If we have in addition that \(R_g\) contains an invertible element of \(R\) for each \(g\in G\) then \(R\) is said to be a \(G\)-crossed product. In [\textit{J. Öinert} and \textit{S. D. Silvestrov}, J. Gen. Lie Theory Appl. 2, No. 4, 287-302 (2008; Zbl 1163.16016)] it was shown that if \(R\) is a \(G\)-crossed product and the subring \(R_e\) is commutative then any non-zero two-sided ideal \(I\) of \(R\) contains at least one non-zero element commuting elementwise with \(R_e\). In the paper under review, this result is generalized to strongly \(G\)-graded rings where \(R_e\) is a commutative ring. Moreover, the authors study the intersection between non-zero two-sided ideals of \(R\) and commutants to bigger subrings than \(R_e\). This is done for strongly graded rings and for a more general class of graded rings to which algebraic crossed products belong, which are called crystaline graded rings. Throughout the paper all rings are assumed to be unital and associative.