In characteristic zero, Zinovy Reichstein and the author generalized the
usual relationship between irreducible Zariski closed subsets of the affine
space, their defining ideals, coordinate rings, and function fields, to a
non-commutative setting, where "varieties" carry a PGL_n-action, regular and
rational "functions" on them are matrix-valued, "coordinate rings" are prime
polynomial identity algebras, and "function fields" are central simple algebras
of degree n. In the present paper, much of this is extended to prime
characteristic.