3
$\begingroup$

Let $R=\oplus_{I\geq 0}R_i$ be a positive graded ring(maybe not commutative), where $R_0$ is a commutative Noetherian ring. If $R$ is finite generated $R_0$-algebra, is $R$ Noetherian?

In here, Is every (left) graded-Noetherian graded ring (left) Noetherian?, $\mathbb Z$-graded ring is graded Noehterian iff it is Noetherian.

I found that this result is true for graded-commutative ring using Artin-Tate lemma:https://en.wikipedia.org/wiki/Artin-Tate_lemma.

Thank you in advance.

$\endgroup$
3
$\begingroup$

The answer is no by Exercice 26 in the 2012 edition of Bourbaki's Algèbre VIII.1. (This seems moreover to have nothing to do with graduations.)

(Translation of the exercise: Let $K$ be a commutative field, let $A$ be the polynomial ring $K[T]$, and let $\sigma$ be the endomorphism $P(T)\mapsto P(T^2)$ of $A$. Then, the ring $A[X]_\sigma$ is not left-noetherian, although $A$ is noetherian.)

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ What is your meaning of graduations? $\endgroup$ – Jian Dec 30 '19 at 7:07
  • $\begingroup$ The usual one, i.e. (for a ring) a given decomposition of the underlying group as a direct sum compatible with the operation of the group of degrees in the obvious way. But the point is that by the linked question it is seen that noetherianness and graded-noetherianness are the same. $\endgroup$ – Fred Rohrer Dec 30 '19 at 8:43
  • $\begingroup$ Hi Rohrer, I don't understand French in the exercise. Can you translate it in the answer? Thank you. $\endgroup$ – Jian Dec 30 '19 at 16:40
  • $\begingroup$ Done. $\mbox{ }$ $\endgroup$ – Fred Rohrer Dec 30 '19 at 18:24
2
$\begingroup$

Perhaps the simplest counterexample: let $R$ be the free $R_0$-algebra on two generators $x$ and $y$. The two-sided ideal $RxR$ is not finitely generated as a left (or right) ideal.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.