Hilbert's Basis Theorem
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R. Formally,
Theorem
If R is a Noetherian ring, then R is a Noetherian ring.
Corollary
If R is a Noetherian ring, then is a Noetherian ring.
For a proof of this result, see the corresponding section on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n variables is generally a geometric object (such as a curve or a surface) in n-space).
Read more about this topic: Noetherian Ring
Famous quotes containing the words basis and/or theorem:
“The self ... might be regarded as a sort of citadel of the mind, fortified without and containing selected treasures within, while love is an undivided share in the rest of the universe. In a healthy mind each contributes to the growth of the other: what we love intensely or for a long time we are likely to bring within the citadel, and to assert as part of ourself. On the other hand, it is only on the basis of a substantial self that a person is capable of progressive sympathy or love.”
—Charles Horton Cooley (1864–1929)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)