Abstract: We generalize 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 particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the ``coordinate ring" of a ``variety'' in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.