On Special J-rings

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))

Literally, “J-ring”. We adopt the modern terminology, “Jordan ring”. [Translators]

Literally, “Jordan rings with an arbitrary ring Σ of operators”. [Translators]