Computing the symmetric ring of quotients
type: Item
instance of: scholarly article
title: Computing the symmetric ring of quotients
zbMATH Open document ID: 0605.16003
DOI: 10.1016/0021-8693(87)90187-6
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let R be an associative prime ring and let \(Q_{\ell}(R)\) be the Martindale ring of quotients [\textit{W. S. Martindale}, J. Algebra 12, 576- 584 (1969; Zbl 0175.031)]. In this paper the author briefly discusses the Martindale ring of quotients, introduces the symmetric Martindale ring of quotients \(Q_ s(R)\) and obtains some basic properties. However, for the most part he computes the symmetric ring of quotients for free algebras and large classes of group algebras. These two problems are posed by S. Montgomery. The prime ring R is symmetrically closed if \(Q_ s(R)=R\). For example, if K is a field and \(R=K<x,y,...>\) is a free K-algebra on at least two generators, then R is symmetrically closed (Theorem 2.5). Or, if every nontrivial normal subgroup of the group G is uncountable, then the group ring K[G] over the field K is symmetrically closed (Theorem 5.6). Let F be a field extension of K. Then K[G] is symmetrically closed if and only if F[G] is symmetrically closed (Theorem 8.1). Furthermore \(Q_{\ell}(K[G])=K[G]\) if and only if \(Q_{\ell}(F[G])=F[G]\) (Theorem 8.2). In fact, most of this paper is devoted to the study of group algebras. Furthermore for the sake of simplicity the author does not always offer the best possible result. But he notes: ''It is clear that much of what we do here extends to crossed products and to more general groups. However, these extensions would necessarily complicate our arguments and are therefore more appropriately put off to a later paper.''
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 3977196
zbMATH Keywords: Martindale ring of quotients
zbMATH Keywords: symmetric Martindale ring of quotients
zbMATH Keywords: free algebras
zbMATH Keywords: group algebras
zbMATH Keywords: prime ring
zbMATH Keywords: crossed products
author: D. S. Passman
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90187-6
OpenAlex ID: W1966644541
cites work: Q5678983
cites work: Prime ideals in differential operator rings and crossed products of infinite groups
cites work: Q3859934
cites work: Semiprime skew group rings
cites work: Maximal quotient rings of group rings
cites work: Ideals in group rings of free products
cites work: Normalizing extensions: Prime ideals and incomparability
cites work: Intermediate Normalizing Extensions
cites work: Q4073508
cites work: Q4180503
cites work: Algebras of invariants of free algebras
cites work: The normal closure of the coproduct of domains over a division ring
cites work: Finite normalizing extensions of rings
cites work: Prime ideals in crossed products of finite groups
cites work: Prime rings satisfying a generalized polynomial identity
cites work: The Normal Closure of the Coproduct of Rings Over a Division Ring
cites work: The normal closure of coproducts of domains
cites work: Automorphism groups of rings with no nilpotent elements
cites work: Crossed products over prime rings
cites work: Q3932957
cites work: Q4742949
cites work: Galois theory of prime rings
cites work: X-inner automorphisms of crossed products and semi-invariants of Hopf algebras
cites work: Observations on group rings
Recommended article: The rings of quotients of free algebras
Recommended article: The maximal symmetric ring of quotients
Recommended article: Functional identities and rings of quotients
Recommended article: The Symmetric Ring Of Quotients Of A Primitive Ring is Primitive
Recommended article: Q3474784
Recommended article: A question of Passman on the symmetric ring of quotients
Recommended article: Q3737588
Recommended article: Partial actions and quotient rings.
Recommended article: Prime nondegenerate Jordan algebras with nonzero socle and the symmetric Martindale algebra of quotients
Recommended article: A general notion of noncommutative Krull rings
description: scientific article; zbMATH DE number 3977196