Polynomial identity rings as rings of functions

Z. Reichstein, N. Vonessen

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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.

Original languageEnglish
Pages (from-to)624-647
Number of pages24
JournalJournal of Algebra
Issue number2
StatePublished - Apr 15 2007


  • Central simple algebra
  • Coordinate ring
  • Nullstellensatz
  • Polynomial identity ring
  • Trace ring


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