## 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.

Original language | English |
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Pages (from-to) | 624-647 |

Number of pages | 24 |

Journal | Journal of Algebra |

Volume | 310 |

Issue number | 2 |

DOIs | |

State | Published - Apr 15 2007 |

## Keywords

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