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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

17C10 Structure theory for Jordan algebras