Semiprimeness of special Jordan algebras
Proc. Amer. Math. Soc. 96 (1986), 29-33
Primary 17C10; Secondary 16A68
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Abstract: There are important connections between radicals of a special Jordan algebra and its associative envelope . For the locally nilpotent (Levitzki) radical , Skosyrskii proved . For the prime (Baer) radical , Erickson and Montgomery proved when consists of all symmetric elements of an algebra with involution . In his important work on prime Jordan algebras, Zelmanov proved for all linear and all associative envelopes . 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.
S. Erickson and S.
Montgomery, The prime radical in special Jordan
rings, Trans. Amer. Math. Soc. 156 (1971), 155–164. MR 0274543
(43 #306), http://dx.doi.org/10.1090/S0002-9947-1971-0274543-4
G. Skosyrskiĭ, Nilpotency in Jordan and right alternative
algebras, Algebra i Logika 18 (1979), no. 1,
73–85, 122–123 (Russian). MR 566775
I. Zel′manov, Prime Jordan algebras. II, Sibirsk. Mat.
Zh. 24 (1983), no. 1, 89–104, 192 (Russian). MR 688595
- T. S. Erickson and M. S. Montgomery, The prime radical in special Jordan rings, Trans. Amer. Math. Soc. 156 (1971), 155-164. MR 0274543 (43:306)
- V. G. Skosyrskiĭ, On nilpotence in Jordan and right alternative algebras, Algebra i Logika 18 (1979), 73-85. MR 566775 (83c:17027)
- E. I. Zelmanov, On prime Jordan algebras. II, Sibirsk Mat. J. 24 (1983), 89-104. MR 688595 (85d:17011)
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