Primitivity in finitely generated special quadratic Jordan algebras
type: Item
instance of: scholarly article
title: Primitivity in finitely generated special quadratic Jordan algebras
zbMATH Open document ID: 0823.17044
author: Fernando Montaner
published in: Journal of Algebra
publication date: 1995-11-09T00:00:00Z
review text: In this interesting paper it is proved that a finitely generated special Jordan algebra \(J\) (over an arbitrary ring of scalars) having a \(*\)- primitive \(*\)-envelope is primitive. The proof of this result is divided into two cases. If \(J\) has nonzero hermitian part then an extension of the Herstein second construction shows that any nonzero ideal of \(J\) contains a nonzero ample ideal. This is the key tool to see that \(J\) is primitive in this case. On the other hand, anti-hermitian finitely generated special Jordan algebras having a \(*\)-primitive \(*\)-envelope are actually finite-dimensional central simple Jordan algebras, and therefore primitive. Since \(*\)-tight envelopes of primitive special Jordan algebras are \(*\)- primitive, the author finally obtains that a finitely generated special Jordan algebra is primitive if and only if any of its \(*\)-tight envelopes is \(*\)-primitive.
Mathematics Subject Classification ID: 17C10
zbMATH DE Number: 703023
zbMATH Keywords: special Jordan algebra
zbMATH Keywords: primitive special Jordan algebras
reviewed by: Antonio Fernández López
MaRDI profile type: Publication
full work available at URL: jabr.1994.1330
OpenAlex ID: W2006401479
DOI: 10.1006/JABR.1994.1330
description: scientific article; zbMATH DE number 703023