Today will just be some quick results we get from this build up.
First, if we localize a polynomial ring at a maximal ideal, say at , then . This is because has Poincare series so the order of the pole is which is the dimension by the last post.
This one will be really useful later: . Let such that are a basis for the vector space. Then by Nakayama’s Lemma the generate . Thus .
This one is also useful in algebraic geometry. If is Noetherian, and , then every minimal ideal belonging to has height . Unfortunately, we cannot push this to equality. Geometrically the example is that if is the twisted cubic, then has height 2, but cannot be generated by less than 3 elements.
Lastly, we’ll do the famous Principal Ideal Theorem. If is Noetherian and is neither a zero-divisor nor a unit, then every minimal prime ideal of has height 1. By the last paragraph we know that . If then it belongs to . Thus every element of is a zero-divisor which is a contradiction since .