Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Semiprimeness of special Jordan algebras

Author: Kevin McCrimmon
Journal: Proc. Amer. Math. Soc. 96 (1986), 29-33
MSC: Primary 17C10; Secondary 16A68
MathSciNet review: 813803
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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)
  • [2] V. G. Skosyrskiĭ, On nilpotence in Jordan and right alternative algebras, Algebra i Logika 18 (1979), 73-85. MR 566775 (83c:17027)
  • [3] E. I. Zelmanov, On prime Jordan algebras. II, Sibirsk Mat. J. 24 (1983), 89-104. MR 688595 (85d:17011)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 17C10, 16A68

Retrieve articles in all journals with MSC: 17C10, 16A68

Additional Information

Keywords: Special Jordan algebra, semiprime, associative enveloping algebra
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society