# 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 – Jian 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.