TY - JOUR

T1 - Polynomial identity rings as rings of functions

AU - Reichstein, Z.

AU - Vonessen, N.

N1 - Funding Information:
* Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Author was supported in part by an NSERC research grant. 2 Author gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

PY - 2007/4/15

Y1 - 2007/4/15

N2 - 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 PGLn-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.

AB - 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 PGLn-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.

KW - Central simple algebra

KW - Coordinate ring

KW - Nullstellensatz

KW - Polynomial identity ring

KW - Trace ring

UR - http://www.scopus.com/inward/record.url?scp=33847684848&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2005.08.008

DO - 10.1016/j.jalgebra.2005.08.008

M3 - Article

AN - SCOPUS:33847684848

SN - 0021-8693

VL - 310

SP - 624

EP - 647

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -