It is proved that Jordan pairs P(n, m) = (Mn. m, Mm,. ) of n X m matrices over a field k are distinguished up to embedding by means of polynomial identities. Also, a basis of identities of P(1, n), where n can be infinite and the characteristic of k is equal to zero, is found. Β© Elsevier Science Inc., 1997
1. Introduction
Polynomial identities are important tools of the structure theory of algebras. Finiteness conditions written in the form of identities are stable under standard algebraic constructions (direct sums, subalgebras, homomorphic images), which makes them very effective in various situations. In this work we study identities of Jordan pairs of rectangular matrices over a field. Note that they form a series of simple pairs [4, p. 196].
We begin with basic notation [4]. Let k be an associative, commutative ring, and let V = (V Γ·, V-) be a pair of k-modules equipped with a pair (Q+, p_) of quadratic mappings Q~: v ~' ~ Hom(V -~, V ~) (or ~ {+, -}). Denote by p~(x, y) the linearization of Q~(x), i.e. Q~(x, y) = Q~(x + y) -Q~(x) -Q~(y); it is a bilinear mapping. For any x, y ~ V a and z ~ V -~ define trilinear mappings V ~ x V -~ x V ~ ~ V ~, (x, y, z) ~ {xyz} and * During 1993/94 at Be,