The universal multiplication envelope UME(J) of a Jordan system J (algebra, triple, or pair) encodes information about its linear actions—all of its possible actions by linear transformations on outer modules M (equivalently, on all larger split null extensions J ⊕ M). In this article, we study all possible actions, linear and nonlinear, on larger systems. This is encoded in the universal polynomial envelope UPE(J), which is a system containing J and a set X of indeterminates. Its elements are generic polynomials in X with coefficients in the system J, and it encodes information about all possible multiplications by J on extensions [image omitted]. The universal multiplication envelope is recovered as the “linear part,” the elements homogeneous of degree 1 in some variable x. We are especially interested in generic polynomial identities, free Jordan polynomials p(x1,…, xn; y1,…, ym) which vanish for particular aj ∈ J and all possible xi in all [image omitted], i.e., such that the generic polynomial p(x1,…, xn; a1,…, am) vanishes in UPE(J). These represent “generic” multiplication relations among elements ai, which will hold no matter where J is imbedded. This will play a role in the problem of imbedding J in a system of “fractions” [image omitted] (McCrimmon, McCrimmon Submitted, McCrimmon To appear). The natural domain for a fraction [image omitted] is the dominion Ks ≻ n = Φ n + Φ s + Us(J) where the denominator s dominates the numerator n in the sense that Un, Un,s are divisible by Us on the left and right. We show that by passing to subdomains we can increase the “fractional” properties of the domain, especially if s generically dominates n in UPE(J).
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