Commutativity and ideals in pre-crystalline graded rings.
type: Item
instance of: scholarly article
title: Commutativity and ideals in pre-crystalline graded rings.
zbMATH Open document ID: 1210.16040
author: Johan Öinert
author: Sergei Silvestrov
published in: Acta Applicandae Mathematicae
publication date: 2010-01-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 13A02
zbMATH DE Number: 5659969
zbMATH Keywords: strongly graded rings
zbMATH Keywords: group graded rings
zbMATH Keywords: commutativity theorems
zbMATH Keywords: ideals
zbMATH Keywords: crossed products
zbMATH Keywords: commutants
zbMATH Keywords: skew group rings
zbMATH Keywords: commutative subrings
reviewed by: María Concepción López-Díaz
MaRDI profile type: Publication
full work available at URL: s10440-009-9434-4
OpenAlex ID: W1992835607
cites work: Q4040687
cites work: Group-Graded Rings, Smash Products, and Group Actions
cites work: Semiprime skew group rings
cites work: Ideals in group rings of free products
cites work: Two-sided ideals in \(q\)-deformed Heisenberg algebras.
cites work: Q4516129
cites work: Q5569204
cites work: Prime ideals of Ore extensions over commutative rings
cites work: Prime ideals of Ore extensions over commutative rings. II
cites work: QUANTUM UNIQUE FACTORISATION DOMAINS
cites work: Primitivity of skew polynomial and skew laurent polynomial rings
cites work: Centers and prime ideals in group algebras of polycyclic-by-finite groups
cites work: Prime ideals in crossed products of finite groups
cites work: Addendum: Prime ideals in crossed products of finite groups
cites work: Graded rings over arithmetical orders
cites work: Q3786586
cites work: Crossed products over prime rings
cites work: Introducing crystalline graded algebras.
cites work: On a correspondence between ideals and commutativity in algebraic crossed products.
cites work: Commutativity and ideals in algebraic crossed products.
cites work: Crossed Product-Like and Pre-Crystalline Graded Rings
cites work: Q4145843
cites work: Some results on the center of a ring with polynomial identity
cites work: DYNAMICAL SYSTEMS AND COMMUTANTS IN CROSSED PRODUCTS
cites work: Q3602122
cites work: Q3769441
cites work: Q4338921
cites work: The interplay between topological dynamics and theory of \(C^*\)-algebra. II: \(C^*\)-algebras and their applications to topological dynamical systems.
cites work: Produits croises d'une \(C^ *\)-algèbre par un groupe d'automorphismes
DOI: 10.1007/S10440-009-9434-4
description: scientific article; zbMATH DE number 5659969