Complementation of inner ideals in Jordan pairs
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.
scientific article; zbMATH DE number 614227