Algebraic Variety - Basic Results

Basic Results

  • An affine algebraic set V is a variety if and only if I(V) is a prime ideal; equivalently, V is a variety if and only if its coordinate ring is an integral domain.
  • Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties (where none of the sets in the decomposition are subsets of each other).
  • Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.

Read more about this topic:  Algebraic Variety

Famous quotes containing the words basic and/or results:

    “A kiss can be a comma, a question mark or an exclamation point. That’s basic spelling that every woman ought to know.”
    —Mistinguett (1874–1956)

    “The study and knowledge of the universe would somehow be lame and defective were no practical results to follow.”
    —Marcus Tullius Cicero (106–43 B.C.)