# Examples of Monoid rings that are quotients of polynomial rings by homogeneous ideals

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?

-
Another example: Laurent polynomials in one variable, obtained as the quotient of the polynomial ring on two generators $x$ and $y$ by the relation $xy=1$. – Yemon Choi Oct 19 '12 at 8:14
$xy=1$ doesn't look homogeneous in $R[x,y]$. – Fernando Muro Oct 19 '12 at 13:45
If by $R[X, X^{-1}]$ you mean a polynomial ring with two indeterminates, then that is not isomorphic to the ring of Laurent polynomials. – user2529 Oct 26 '12 at 6:19
By $A=R[X^{\pm 1}]$ I mean the ring $R[X,Y]/(XY-1)$. Clearly, $A$ is the ring of Laurent polynomials and it is also isomorphic to the group ring $R[\mathbb Z]$. There are other ways to see this ring that are connected with the view-point of monoid rings. In fact, if you take the monoid ring $R[\mathbb N]\cong R[X]$, and you consider the multiplicative system $\Sigma_X=${$X,X^2,\dots,X^n,\dots$} of central elements in $\Sigma$, then the ring $R[\mathbb Z]\cong R[X^{\pm 1}]$ is the Ore localization $\Sigma^{-1}R[X]$. – Simone Virili Oct 28 '12 at 23:06
Thank you very much for your clarification, Simone. – user2529 Nov 13 '12 at 14:01