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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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