Ascending / descending chain condition on graded modules.

Question

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.

Answer 1

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.