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Characterization of the McCrimmon radical. (Russian) Zbl 0559.17001

It is considered the Jordan triple system (J.t.s.) T over an associative commutative ring \(\Phi\) \(\ni\), i.e. a \(\Phi\)-module with trilinear operation \(\{\) x,y,z\(\}\), satisfying the following identities \[ (1)\quad \{x,y,\{x,z,x\}\}=\{x,\{y,x,z\},x\}; \]
\[ (2)\quad \{\{x,y,x\},y,z\}=\{x,\{y,x,y\},z\}; \]
\[ (3)\quad \{\{x,y,x\},z,\{x,y,x\}\}=\{\{x,\{y,\{x,z,x\},y\},x\} \] and all their linearizations. The element \(a\in T\) is called an absolute zero divisor if \(\{a,T,a\}=0\). We call a J.t.s. nondegenerate if T does not contain absolute zero divisors. The minimal ideal \({\mathfrak M}(T)\) for which the quotient system T/\({\mathfrak M}(T)\) is nondegenerate is called McCrimmon ideal of T. The sequence \(\{x_ n| n\geq 1\}\) of elements of a J.t.s. is called an m-system if \(x_{n+1}\in \{x_ n,T,x_ n\}\) for \(n\geq 1\). The J.t.s. T is called prime if for any non-zero ideals \(K,L\triangleleft T\) the ideal \(K\#L=\{K,L,T\}+\{L,K,T\}+\{K,T,L\}\) is also non-zero.
The basic results are the following. Theorem 1. \({\mathfrak M}(T)\) coincides with the set of all m-elements of T. - Theorem 2. Any subsystem \(A\leq {\mathfrak M}(T)\) is radical in the sense of McCrimmon. - Theorem 3. \({\mathfrak M}(T)=\cap_{P\in {\mathcal P}}P\), where \({\mathcal P}\) denotes the set of all ideals \(I\triangleleft T\) for which the quotient-system T/I is prime and non-degenerate.
Reviewer: A.Fleischer

MSC:

17A40 Ternary compositions
17C50 Jordan structures associated with other structures
17C99 Jordan algebras (algebras, triples and pairs)
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