Noetherian Ring - Uses

Uses

The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension.