On invariant additive subgroups
type: Item
instance of: scholarly article
title: On invariant additive subgroups
zbMATH Open document ID: 0618.16002
DOI: 10.1007/BF02769465
author: B. George
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The author obtains the most comprehensive structure theorem to date for additive subgroups of prime rings which are invariant under certain kinds of automorphisms. Specifically, let A be an additive subgroup of the prime ring R and assume that for every \(x\in R\) satisfying \(x^ 2=0\), \((1+x)A(1-x)\subset A\). If Q is the Martindale quotient ring of R, let \(N=\{q\in Q|\) \(Iq+qI\subset R\) for some nonzero ideal I of \(R\}\). The main theorem asserts that when N contains a nontrivial idempotent then either A is central, A contains a noncommutative Lie ideal of R, or char R\(=2\) and R satisfies the standard identity \(S_ 4\). The author proceeds to analyze the last possibility and determines a condition on the center of R which is equivalent to the statement that a noncentral A must contain a noncentral Lie ideal of R. The main theorem is the additive version of [\textit{I. N. Herstein}, J. Algebra 83, 26-32 (1983; Zbl 0514.16001)], and its proof both uses the results in and follows the general approach of this paper of Herstein. The importance of the main theorem here lies in the fact that it is essentially the first result for subgroups and it implies and genuinely extends almost all of the previous results in this area, which before Herstein's paper, considered only invariant subspaces or invariant subalgebras. The complications resulting from the restriction to additive subgroups are nontrivial, and the main problem is overcome in Lemma 4 with a very clever and nice computation.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16P50
zbMATH DE Number: 4002294
zbMATH Keywords: additive subgroups of prime rings
zbMATH Keywords: automorphisms
zbMATH Keywords: Martindale quotient ring
zbMATH Keywords: Lie ideal
zbMATH Keywords: standard identity
zbMATH Keywords: center
MaRDI profile type: Publication
cites work: Invariant Submodules of Simple Rings
cites work: Q5639810
cites work: Q4112071
cites work: A theorem on invariant subrings
cites work: Differential identities of prime rings
cites work: Lie structure of prime rings of characteristic 2
cites work: Prime rings satisfying a generalized polynomial identity
cites work: Derivations in Prime Rings
cites work: Q3249822
full work available at URL: https://doi.org/10.1007/bf02769465
OpenAlex ID: W1991904604
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description: scientific article; zbMATH DE number 4002294