Example of a graded module that is not a ring.

Question

I'm looking for an example of a graded module, that is not a ring.

All the examples of graded modules that I have come across, like $k[x_1,x_2,\dots,x_n]$ are all graded rings.

Thanks in advance

Answer 1

Take any ring $R$ and any $R$-module $M$. Then $R$ is a graded ring with graduation concentrated in degree $0$ (ie $R_0=R$ and $R_n = 0$ if $n\neq 0$), and $M$ is a graded module also with graduation concentrated in degree $0$.

If your $R$-module $M$ was not a ring, the graded version is not a ring either.

Basically, what I'm saying is that you can see ordinary rings and modules as trivially graded objects, with no cost at all, and the behaviour will be the classical one.

Answer 2

Let $P=k[x_1,\ldots,x_n]$ be a polynomial ring and let $G$ be a group acting nontrivially on $P$. Let $R$ be the skew group ring of $G$ over $P$, which is the free left $P$-module with multiplication defined by $$pg\cdot qh=p(g\cdot q)gh$$ for $p,q\in P$, $g,h\in G$. This is a graded ring where the elements of $G$ have degree 0 and the elements of $P$ have their usual degrees.

Now the tensor product of left modules $R\otimes_P R$ over $P$ is not a ring, but it is naturally a graded module over $R$, where $$pg\cdot (r\otimes s)=p(gr\otimes gs)$$