Generic 2 × 2 matrices in positive characteristic

Let Rm(K) be the K-algebra generated by the generic 2 × 2 matrices y1, …, ym over a unitary commutative associative ring K. Our main result is that, with a small class of exceptions, for m a positive integer and p a prime, the kernel of the natural homomorphism Rm( ) → Rm( p) coincides with pRm( ). The only exceptions are for m ≥ 5 and p = 2 when we give an explicit multilinear polynomial identity of degree 5 for the matrix algebra M2( 2) which does not follow from the spolynomial identities of M2( 2). This improves on Schelterʹs construction of a non-multilinear identity of this sort of degree 6, and Drensky and Tsiganchevʹs existence result for a multilinear identity such as we have found