Order relation in quadratic Jordan rings and a structure theorem

González, Santos; Martínez, Consuelo

doi:10.1090/S0002-9939-1988-0958042-8

Order relation in quadratic Jordan rings and a structure theorem
HTML articles powered by AMS MathViewer

by Santos González and Consuelo Martínez PDF

Proc. Amer. Math. Soc. 104 (1988), 51-54 Request permission

Abstract:

It is shown that the relation defined by $x \leq y$ if and only if ${V_x}x = {V_x}y$ and ${U_x}x = {U_x}y = {U_y}x$ is an order relation for quadratic Jordan algebras without nilpotent elements, which extends our previous one for linear Jordan algebras, and reduces to the usual Abian order for associative algebras. We prove that a quadratic Jordan algebra is isomorphic to a direct product of division algebras if and only if the algebra has no nilpotent elements and is hyperatomic and orthogonally complete.

References

Alexander Abian, Direct product decomposition of commutative semi-simple rings, Proc. Amer. Math. Soc. 24 (1970), 502–507. MR 258815, DOI 10.1090/S0002-9939-1970-0258815-X
Alexander Abian, Order in a special class of rings and a structure theorem, Proc. Amer. Math. Soc. 52 (1975), 45–49. MR 374222, DOI 10.1090/S0002-9939-1975-0374222-8
Alexander Abian, Addendum to “Order in a special class of rings and a structure theorem” (Proc. Amer. Math. Soc. 52 (1975), 45–49), Proc. Amer. Math. Soc. 61 (1976), no. 1, 188 (1977). MR 419548, DOI 10.1090/S0002-9939-1976-0419548-5
M. Chacron, Direct product of division rings and a paper of Abian, Proc. Amer. Math. Soc. 29 (1971), 259–262. MR 274512, DOI 10.1090/S0002-9939-1971-0274512-X
Santos González and Consuelo Martínez, Order relation in Jordan rings and a structure theorem, Proc. Amer. Math. Soc. 98 (1986), no. 3, 379–388. MR 857926, DOI 10.1090/S0002-9939-1986-0857926-7
S. González and C. Martínez, Order relation in JB-algebras, Comm. Algebra 15 (1987), no. 9, 1869–1876. MR 898297, DOI 10.1080/00927878708823509
N. Jacobson, Lectures on quadratic Jordan algebras, Tata Institute of Fundamental Research Lectures on Mathematics, No. 45, Tata Institute of Fundamental Research, Bombay, 1969. MR 0325715

Structure theory of Jordan algebras

Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721, DOI 10.1007/BFb0080843
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
Hyo Chul Myung and Luis R. Jimenez, Direct product decomposition of alternative rings, Proc. Amer. Math. Soc. 47 (1975), 53–60. MR 354796, DOI 10.1090/S0002-9939-1975-0354796-3