Commutativity and ideals in pre-crystalline graded rings.
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{E. Nauwelaerts} and \textit{F. Van Oystaeyen}, Algebr. Represent. Theory 11, No. 2, 133-148 (2008; Zbl 1163.16033)] a new class of graded rings was introduced which are called (pre-) crystalline graded rings. This is a generalization of algebraic crossed products. In the paper under review, the authors describe the center and the commutant of the degree zero grading part in general pre-crystalline graded rings. The authors consider the following assertions: (1) the ring \(R_e\) is a maximal commutative subring in \(R\), and (2) Every non-zero two-sided ideal in \(R\) contains a non-zero element in \(R_e\). For crystalline graded rings and skew group rings, they provide sufficient conditions on \(R_e\), the grading group \(G\), the cocycle and the action for equivalence between (1) and (2). An example of a situation where the equivalence does not hold is given, showing that (1) may be false even though (2) is true. In the cases where \(R_e\) is commutative, sufficient conditions for (1) to imply (2) are obtained. Throughout the paper all rings are assumed to be unital and associative.
scientific article; zbMATH DE number 5659969
scientific article