Noetherian Rings

We say that a Ring is Noetherian
if any ascending chain of Ideals eventually terminates.

Lemma

A Ring is Noetherian
if and only if
every Ideal is finitely generated.

Hilbert basis theorem

If is Noetherian, then is Noetherian.

Proof

Take an ideal of and suppose it is not finitely generated.
Find of minimal degree in .
Then find of minimal degree.
Take their leading coefs
and note that the sequence of ideals must terminate
i.e. .
Hence find a polynomial
with same degree and same leading coef as .
Now
but then wasn’t of minimal degree - contradiction.
So any is finitely generated,
hence is Noetherian.

Lemma

Any Quotient Ring of a Noetherian ring is Noetherian.