# Ascending / descending chain condition on graded modules.

Let $$R = \bigoplus_{n \in \mathbb{N}} R_n$$ be a graded commutative ring. Then $$R$$ is noetherian / artinian if and only if it has the ascending / descending chain condition for homogeneous ideals, see Ascending chain conditions on homogeneous ideals.

Is the same also true for a graded module $$M = \bigoplus_{n\in \mathbb{Z}} M_n$$? I.e. is $$M$$ noetherian / artinian if and only if it has the ascending / descending chain condition for graded submodules?

I actually only want to use this if $$R$$ is noetherian, but I'm not sure this helps in any way.

For some context why I'm interested in this, see Divided power algebra is artinian as a module over the polynomial ring.

• well, if I remember my algebra correctly, for finitely generated modules, these properties should literally just carry over. i.e. $R$ is noetherian (artinian) if and only if the analogous chain conditions hold for all finitely generated modules. – Enkidu Oct 31 '18 at 13:32
• The ring is positive graded ring in the proof of.in math.stackexchange.com/q/147466/453628 Is this true for atbitrary graded ring – Sky Oct 31 '18 at 23:56
• @Enkidu Yes, this is easy, because finitely generated modules are quotients of some $R^n$ and being noetherian/artinian is preserved by finite sums and quotients. But In my application, $R$ is not artinian and $M$ is not finitely generated either, see my other linked question. I would like to know if it would suffice to show the acc/dcc for graded submodules of $M$ only. – red_trumpet Nov 1 '18 at 7:42
• @Sky In my application the ring itself is positively graded, but the module is not. I don't know about the other cases. – red_trumpet Nov 1 '18 at 7:44

A $$\mathbb{Z}$$-graded module over an $$\mathbb{Z}$$-graded ring is noetherian as a graded module if and only if it is so as an ungraded module.

A $$\mathbb{Z}$$-graded module with well-ordered support over an $$\mathbb{Z}$$-graded ring is artinian as a graded module if and only if it is so as an ungraded module.

This holds by Propositions 5.4.7 and 5.4.5 in C. Nastasescu, F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics 1836, Springer, Berlin, 2004.

I do not know whether the hypothesis on the support in the second statement can be omitted, but I doubt it.