Order relation in quadratic Jordan rings and a structure theorem

Authors:
Santos González and Consuelo Martínez

Journal:
Proc. Amer. Math. Soc. **104** (1988), 51-54

MSC:
Primary 17C10

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958042-8

MathSciNet review:
958042

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the relation defined by if and only if and 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.

**[1]**A. Abian,*Direct product decomposition of commutative semisimple rings*, Proc. Amer. Math. Soc.**24**(1970), 502-507. MR**0258815 (41:3461)****[2]**-,*Order in a special class of rings and a structure theorem*, Proc. Amer. Math. Soc.**52**(1975), 45-49. MR**0374222 (51:10422)****[3]**-,*Addendum to "Order in a special class of rings and a structure theorem*", Proc. Amer. Math. Soc.**61**(1976), 188. MR**0419548 (54:7569)****[4]**M. Chacron,*Direct product of division rings and a paper of Abian*, Proc. Amer. Math. Soc.**29**(1971), 259-262. MR**0274512 (43:275)****[5]**S. González and C. Martínez,*Order relation in Jordan rings and a structure theorem*, Proc. Amer. Math. Soc.**98**(1986), 379-388. MR**857926 (88a:17041)****[6]**-,*Order relation in**-algebras*, Comm. Algebra**15**(1987), 1869-1877. MR**898297 (88h:46127)****[7]**N. Jacobson,*Lectures on quadratic Jordan algebras*, Lecture Notes, Tata Institute, Bombay, 1969. MR**0325715 (48:4062)****[8]**-,*Structure theory of Jordan algebras*, Lecture Notes in Math., University of Arkansas, 1971.**[9]**O. Loos,*Jordan pairs*, Lecture Notes in Math., vol. 460, Springer-Verlag, Berlin and New York, 1975. MR**0444721 (56:3071)****[10]**K. McCrimmon,*A general theory of Jordan rings*, Proc. Nat. Acad. Sci. U.S.A.**56**(1966), 1072-1079. MR**0202783 (34:2643)****[11]**H. C. Myung and L. Jimenez,*Direct product decomposition of alternative rings*, Proc. Amer. Math. Soc.**47**(1975), 53-60. MR**0354796 (50:7273)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958042-8

Keywords:
Quadratic Jordan algebras,
partial order,
direct product,
nilpotent element,
hyperatomic,
orthogonally complete

Article copyright:
© Copyright 1988
American Mathematical Society