  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. In addition, a mistake in the earlier paper is corrected. One
of the results is that the finitely generated prime PI-algebras of degree n are
precisely the rings that arise as "coordinate rings" of "n-varieties" in this
setting. For n = 1 the definitions and results reduce to those of classical
affine algebraic geometry.
