Generic 2 x 2 matrices in positive characteristic

Let R-m(K) be the K-algebra generated by the generic 2 x 2 matrices y(1),.. ., y(m) over a unitary commutative associative ring K. Our main result is t hat, with a Small class of exceptions, for m a positive integer and p a pri me, the kernel of the natural homomorphism R-m(Z) --> R-m(Z(p)) coincides w ith pR(m)(Z). The only exceptions are for m greater than or equal to 5 and p = 2 when we give an explicit multilinear polynomial identity of degree 5 for the matrix algebra M-2(Z(2)) which does not follow from the polynomial identities of M-2(Z). This improves on Schelter's construction of a non-mul tilinear identity of this sort of degree 6, and Drensky and Tsiganchev's ex istence result for a multilinear identity such as we have found. (C) 2000 A cademic Press.