I am looking for examples of monoid rings $R(M)$ that is a quotient $R[X_1,\ldots]/{\mathfrak a}$ of a polynomial ring $R[X_1,\ldots ]$ which any number of indeterminates and ${\mathfrak a}$ is a homogeneous ideals.

I have two examples. First is $R({\mathbb N})$ where ${\mathbb N}$ is the natural numbers under addition. Then $R({\mathbb N})\cong R[X]$.

Let $p$ be a prime. Another example is ${\mathbb F}_p({\mathbb Z}/p)$ where ${\mathbb F}_p$ is the finite field with $p$ elements and ${\mathbb Z}/p$ is the cyclic group of integers mod $p$. Then $${\mathbb F}_p({\mathbb Z}/p) \cong {\mathbb F}_p[\bar{t}]/(\bar{t}^p)$$ where $t$ is a generator of ${\mathbb F}_p$ and $\bar{t}=t-1$.

Do you have other examples or general sufficient conditions?