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:

    “There’s a basic rule which runs through all kinds of music, kind of an unwritten rule. I don’t know what it is. But I’ve got it.”
    —Ron Wood (b. 1947)

    “The restlessness that comes upon girls upon summer evenings results in lasting trouble unless it is speedily controlled. The right kind of man does not look for a wife on the streets, and the right kind of girl waits till the man comes to her home for her.”
    —Sedalia Times (1900)