Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: *is it true that $\sqrt{I}$ is homogeneous*? In this generality, the answer is no (see comments). However, if $M$ is a cancellative monoid with a total order, then the usual proof for $M=\mathbb{N}$ works, and indeed $\sqrt{I}$ is homogeneous. So:

- Is there a natural class of monoids $M$ (larger, or different, from totally ordered cancellative monoids) such that in every $M$-graded algebra the radical of a homogeneous ideal is homogeneous?
- The same question, for fixed $R$. In darij grinberg's example, it is relevant that the characteristic of the ring is 2. So, given a ring $R$, is there a natural class of monoids such that in every $M$-graded $R$-algebra the radical of a homogeneous ideal is homogeneous?

I am assuming everything in sight is commutative.