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Goldie’s theorem for Jordan algebras. (Russian) Zbl 0638.17012
Let J be a Jordan algebra over a field of characteristic $$\neq 2$$. For a,b$$\in J$$, D(a,b) denotes the inner derivation which maps each x to (xa)b-(xb)a; and $$Ann_ J(M)=\{a\in J$$; $$Ma=JD(M,a)=0\}$$ is called the annihilator of the subset $$M\subset J$$. Let $$K_ J(M)$$ denote the smallest inner ideal of J containing M. Then a collection $$\{Q_ i\}$$ of inner ideals in J is said to form a direct sum if $$Q_ i\cap K_ J(Q_ j: j\neq i)=0$$ for all $$i\geq 1$$. The algebra J is called a Goldie algebra if it (1) satisfies the ascending (or, equivalently, descending) chain condition on annihilators, and (2) does not contain infinite direct sums of inner ideals.
Setting $$\{x,y,z\}=(xy)z+x(yz)-y(xz)$$, let $$S\subseteq J$$ be such that $$s_ 1,s_ 2\in S$$ implies $$\{s_ 1,s_ 2,s_ 1\}\in S$$. Then J is said to be an order with respect to S in the algebra $$Q\supseteq J$$ if (1) every element from S is invertible in Q; (2) for every $$r\in J$$ there is an $$s\in S$$ such that $$r\cdot s\in J$$ and JD(r,s)$$\subseteq J$$; $$(3)\quad \{a,S,a\}\cap \{b,S,b\}\neq \emptyset$$ for any a,b$$\in S.$$
The author’s main result is now the following: a prime (semiprime) Jordan Goldie algebra is an order in a simple (semisimple) Jordan algebra with minimal condition on inner ideals.
Reviewer: H.F.Smith

##### MSC:
 17C10 Structure theory for Jordan algebras