Complementation of inner ideals in Jordan pairs
type: Item
instance of: scholarly article
title: Complementation of inner ideals in Jordan pairs
zbMATH Open document ID: 0827.17032
author: Ottmar Loos
author: Erhard Neher
published in: Journal of Algebra
publication date: 1994-08-17T00:00:00Z
review text: The well-known module-theoretic characterization of a semiprime left Artinian associative ring \(R\) consists of the following condition on \(R\): \(R\) is unital and completely reducible as a left \(R\)-module. The authors establish a Jordan analogue of this characterization. The pivotal concept introduced here is that of a complement of inner ideal in a Jordan pair. Let \(V= (V^+, V^-)\) be a Jordan pair and \(M\subset V^+\) an inner ideal. With \(M\) the authors associate its kernel \(\text{Ker } M= \{y\in V^-\mid Q_M y= Q_M Q_y M=0\}\), a subspace of \(V^-\). Then a complement of \(M\) is an inner ideal \(N\subset V^-\) such that \(V^+= M\oplus \text{Ker } N\), \(V^-= N\oplus \text{Ker } M\). Similarly, a complement of an inner ideal of \(V^-\) is defined. A nondegenerate Jordan pair is called complemented if every inner ideal both in \(V^+\) and in \(V^-\) admits a complement. This proves to be the appropriate Jordan version of complete reducibility, as the following main results by the authors show: i) A Jordan pair is complemented if and only if it is a direct sum of simple complemented Jordan pairs; ii) a simple Jordan pair is complemented if and only if it is Artinian and nondegenerate. These results, together with the classification of simple Artinian Jordan pairs given in \textit{O. Loos}' book [Jordan pairs (Lect. Notes Math. 460) (Springer 1975; Zbl 0301.17003)], constitute the Jordan pair analogue of the classical structure theorems for semiprime Artinian rings.
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 17C20
zbMATH DE Number: 614227
zbMATH Keywords: Artinian Jordan pair
zbMATH Keywords: complement of inner ideal
zbMATH Keywords: Jordan pair
reviewed by: Ivan Shestakov
MaRDI profile type: Publication
full work available at URL: jabr.1994.1151
OpenAlex ID: W1992203768
DOI: 10.1006/JABR.1994.1151
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description: scientific article; zbMATH DE number 614227