## Is the radical of a homogeneous ideal homogeneous?

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 $M=\mathbb{N}$ … Read more