The University of Montana

Department of Mathematical Sciences

Technical report #30/2010

Polynomial Identity Rings as Rings of Functions, II

Nikolaus Vonessen

**Abstract**

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.
**Keywords:** Polynomial identity ring, central simple algebra, trace ring,
coordinate ring, the Nullstellensatz, n-variety.

**2010 Mathematics Subject Classification.** Primary: 16R30, 16R20; Secondary 14L30,
14A10.

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