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The Andrunakievich lemma and Jordan algebras

Throughout, Λ is an associative commutative ring with 1 and an invertible 2, and J is a Jordan algebra. In the general theory of radicals of associative algebras an important role is played by Andrunakievich's lemma. One of the important problems of the general theory of radicals is to establish analogues of this lemma in different varieties of algebras. A variant of Andrunakievich's lemma for Jordan algebras was established by A.M. Slin'ko consisting in the following: if I is an ideal of the algebra J, Μ is an ideal of the algebra /, and the quotient algebra I/M has no non-zero nilpotent ideals, then Μ is an ideal of J ([2], Ch. 14, §4, Theorem 12). As experience shows, the sphere of application of this theorem is restricted to the class of hypernilpotent radicals (that is, radicals containing all the nilpotent algebras). In this paper we use the line of argument in the proof of [2], Ch. 14, §4, Lemma 17 to prove a somewhat stronger version of Andrunakievich's lemma for Jordan algebras, which we then apply to the study of arbitrary radicals of Jordan algebras (see [1], [4]). As Yu.A. Medvedev has shown, the exact analogue of Andrunakievich's lemma for Jordan algebras is false [3]. We note also that Proposition 1 fills a gap in the proof of [1], Propositon 1.1 and removes the ambiguities in the proof of [4], Theorem 2. Propositon Let