Is the monoid ring of a Noetherian monoid Noetherian?

Question

Let $M$ be a monoid. Suppose that $M$ is left Noetherian, i.e. that every increasing chain of left ideals in $M$ stabilizes. Then is the monoid ring $\mathbb{C}[M]$ necessarily a left Noetherian ring? If so, is it still true if $\mathbb{C}$ is replaced by an arbitrary left Noetherian ring?

Answer 1

No, that would imply that all group algebras over fields are Noetherian. Consider the following.

It is well-known (see Connell, I. G., On the group ring, Can. J. Math. 15, 650-685 (1963). ZBL0121.03502) that the group ring $R[G]$ being right Noetherian entails that $G$ has the maximum condition on subgroups (and that $R$ is right Noetherian.) So if you take any group without the maximum condition on subgroups, you have a monoid with only trivial left/right ideals, and $R[G]$ cannot be Noetherian. Of course, this all holds with $R=\mathbb C$.

While I was probing for further research, I ran across this which you may want to look at:

Rush, David E., Noetherian properties in monoid rings., J. Pure Appl. Algebra 185, No. 1-3, 259-278 (2003). ZBL1084.13007.