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:

    “We can’t nourish our children if we don’t nourish ourselves.... Parents who manage to stay married, sane, and connected to each other share one basic characteristic: The ability to protect even small amounts of time together no matter what else is going on in their lives.”
    —Ron Taffel (20th century)

    “A distinction of property results from that very protection which a free Government gives to unequal faculties of acquiring it.”
    —James Madison (1751–1836)