Siamack Bondari
Abstract
Suppose that R = (R, +, ) is a nonassociative ring, and that R+ = (R, +, ˆ) is a Jordan ring where the ˆ-product is defined by x ˆ y = xy + yx. We compute the minimal degree three and degree four identities which are strong enough to make R third-power associative, R+ Jordan, and the McCrimmon radical of R+ to be an ideal of R. We display an absorption identity of R which demonstrates that the McCrimmon radical of R+ is an ideal of R. Because of the sizes of the matrices involved, the work was done modulo various primes and extrapolated to the rational numbers. The goal was to present the absorption identities. Actual proofs are too large to publish. The interested reader may check them using computer algebra packages. For (R+, +, ˆ) to have a McCrimmon radical, we have to have characteristic ≠ 2.The heart of the paper is a solution to the matrix equation AX = B where both the matrix A and the vector B have entries which are rational functions in α. The solution proceeds by solving the system modulo various primes for various values of α and using these residues to estimate the vector of rational functions X. A genetic algorithm is used to maximize the number of zero entries in the solution X.
Index Terms
- The McCrimmon radical for identities of degree 3
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