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Radicals in Jordan algebras. (English. Russian original) Zbl 0662.17015
Sib. Math. J. 29, No. 2, 283-293 (1988); translation from Sib. Mat. Zh. 29, No. 2(168), 154-166 (1988).
Let \(J\) be a Jordan \(\Phi\)-algebra, where \(\Phi\) is a commutative associative ring with \(1/2\); and let \(R(J)\) denote the multiplicative enveloping algebra of \(J\). The author first proves:
(1) For \(Z\) an ideal of \(\Phi\), let \({\mathcal L}_Z(J)\) and \(\mathcal L_Z(R(J))\) be the locally finite over \(Z\) radicals of \(J\) and \(R(J)\), respectively. Then \(U(\mathcal L_Z(J))\subseteq \mathcal L_Z(R(J))\).
(2) Let \(\mathcal L(J)\) and \(\mathcal L(R(J))\) be the locally nilpotent radicals of \(J\) and \(R(J)\), respectively. Then \((\mathcal L(J))^\tau\subseteq \mathcal L(R(J))\), where \(\tau\) maps \(a\in J\) to \(R_a\in R(J).\)
These two results are then used to show the following:
(3) Let \(M\) be an irreducible bimodule of the \(\Phi\)-algebra \(J\), and \(I\) be an ideal of \(J\). Then either \(M\) is an irreducible \(I\)-bimodule, or \(U(I)\subseteq \operatorname{Ann}(M) = \{W\in R(J)\mid MW=0\}\). In particular, if \(I\) is a minimal ideal of \(J\), then either \(I^2=0\) or \(I\) is a simple algebra.
(4) Let \(B\) be an ideal of \(J\) such that for some integer \(n\) the \(\Phi\)-algebra \(J\) satisfies the minimal condition for ideals contained in \(B^n\). If \(B\) is locally nilpotent, or if \(B\) is contained in the antisimple radical of \(J\), then \(B\) is nilpotent.

MSC:
17C10 Structure theory for Jordan algebras
17C17 Radicals in Jordan algebras
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References:
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