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On Special J-rings

  • A. I. Shirshov
Part of the Contemporary Mathematicians book series (CM)

Abstract

A commutative ring such that for every pair of elements a and b the following equation holds,
$$ J_0 \{ a,b\} \equiv (a^2 b)a - a^2 (ba) = 0, $$
(1)
is called a Jordan ring 1. In the first four sections of this paper, we will consider Jordan algebras2 over an arbitrary ring of coefficients Σ, assuming only that Σ is a unital ring and that for each element a in the Jordan algebra there exists a unique element b such that 2b=a. Clearly, in this case the equation 2a=0 implies a=0. In such Jordan algebras, i.e., Jordan algebras without elements of order 2 in the additive group, the following equations hold:
$$ \begin{gathered} J_1 \{ x,y,z,t\} \equiv \hfill \\ [(yz)x]t + [(ty)x]z + [(zt)x]y - (yz)(xt) - (ty)(xz) - (zt)(xy) = \hfill \\ \end{gathered} $$
(2)
$$\begin{gathered} 0, \hfill \\ J_2 \{ x,y,z,t\} \equiv \hfill \\ [(yz)x]t + [(ty)x]z + [(zt)x]y - (xz)y]t - [(tx)y]z - [(zt)y]x = 0. \hfill \\ \end{gathered} $$
(3)

Keywords

Natural Number Inductive Hypothesis Additive Group Associative Algebra Jordan Algebra 
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References

  1. [1]
    A.A. Albert, A note on the exceptional Jordan algebra, Proc. Nat. Acad. Sci. U.S.A. 36 (1950) 372–374.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P.M. Cohn, On homomorphic images of special Jordan algebras Canadian J. Math. 6 (1954) 253–264.MATHMathSciNetGoogle Scholar
  3. [3]
    A.I. Malcev, On a representation of nonassociative rings, Uspekhi Mat. Nauk N.S. 7 (1952) 181–185.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2009

Authors and Affiliations

  • A. I. Shirshov

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