Jordan rings with involution

Ng, Seong Nam

doi:10.1090/S0002-9947-1974-0399198-2

Jordan rings with involution
HTML articles powered by AMS MathViewer

by Seong Nam Ng PDF

Trans. Amer. Math. Soc. 200 (1974), 111-139 Request permission

Abstract:

Let $J$ be a Jordan ring with involution $\ast$ in which $2x = 0$ implies $x = 0$ and in which $2J = J$. Let the set $S$ of symmetric elements of $J$ be periodic and let $N$ be the Jacobson radical of $J$. Then ${N^2} = 0$ and $J/N$ is a subdirect sum of $\ast$-simple Jordan rings of the following types (1) a periodic field, (2) a direct sum of two simple periodic Jordan rings with exchange involution, (3) a $3 \times 3$ or $4 \times 4$ Jordan matrix algebra over a periodic field, (4) a Jordan algebra of a nondegenerate symmetric bilinear form on a vector space over a periodic field.

References

Goldie-like conditions on Jordan matrix rings

Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
Kevin McCrimmon, A general theory of Jordan rings, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1072–1079. MR 202783, DOI 10.1073/pnas.56.4.1072
Kevin McCrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 671–678. MR 268238, DOI 10.1073/pnas.62.3.671
Kevin McCrimmon, A characterization of the radical of a Jordan algebra, J. Algebra 18 (1971), 103–111. MR 277583, DOI 10.1016/0021-8693(71)90129-3
Susan Montgomery, A generalization of a theorem of Jacobson, Proc. Amer. Math. Soc. 28 (1971), 366–370. MR 276272, DOI 10.1090/S0002-9939-1971-0276272-5
J. Marshall Osborn, Jordan algebras of capacity two, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 582–588. MR 215892, DOI 10.1073/pnas.57.3.582
J. Marshall Osborn, Jordan and associative rings with nilpotent and invertible elements, J. Algebra 15 (1970), 301–308. MR 262316, DOI 10.1016/0021-8693(70)90058-X
J. Marshall Osborn, Varieties of algebras, Advances in Math. 8 (1972), 163–369 (1972). MR 289587, DOI 10.1016/0001-8708(72)90003-5
Chester Tsai, The prime radical in a Jordan ring, Proc. Amer. Math. Soc. 19 (1968), 1171–1175. MR 230776, DOI 10.1090/S0002-9939-1968-0230776-X