Semiprimeness of special Jordan algebras
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- by Kevin McCrimmon PDF
- Proc. Amer. Math. Soc. 96 (1986), 29-33 Request permission
Abstract:
There are important connections between radicals of a special Jordan algebra $J$ and its associative envelope $A$. For the locally nilpotent (Levitzki) radical $\mathcal {L}$, Skosyrskii proved $\mathcal {L}(J) = J \cap \mathcal {L}(A)$. For the prime (Baer) radical $\mathcal {P}$, Erickson and Montgomery proved $\mathcal {P}(J) = J \cap \mathcal {P}(A)$ when $J = H(A, * )$ consists of all symmetric elements of an algebra $A$ with involution $*$. In his important work on prime Jordan algebras, Zelmanov proved $\mathcal {P}(J) = J \cap \mathcal {P}(A)$ for all linear $J$ and all associative envelopes $A$. In the present paper we extend Zelmanov’s result to arbitrary quadratic Jordan algebras. In particular, we see that a special Jordan algebra is semiprime iff it has some semiprime associative envelope.References
- T. S. Erickson and S. Montgomery, The prime radical in special Jordan rings, Trans. Amer. Math. Soc. 156 (1971), 155–164. MR 274543, DOI 10.1090/S0002-9947-1971-0274543-4
- V. G. Skosyrskiĭ, Nilpotency in Jordan and right alternative algebras, Algebra i Logika 18 (1979), no. 1, 73–85, 122–123 (Russian). MR 566775
- E. I. Zel′manov, Prime Jordan algebras. II, Sibirsk. Mat. Zh. 24 (1983), no. 1, 89–104, 192 (Russian). MR 688595
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 29-33
- MSC: Primary 17C10; Secondary 16A68
- DOI: https://doi.org/10.1090/S0002-9939-1986-0813803-9
- MathSciNet review: 813803