Is $\mathbb{Z}[x,y,z]$ an $\mathbb{N}$-graded ring?

Question

I know that my question seems to be obvious, but I really need the answer

Let $\mathbb{Z}[x, y, z]$ be a polynomial ring. I know that it is a $\mathbb{Z}$-graded ring, but is it not in fact $\mathbb{N}$-graded? I don't see why it is not $\mathbb{N}$-graded if we consider the degrees of polynomials. But when I tried to use Nakayama's Lemma (the graded version), my professor told me that we cannot use it for any polynomial ring with coefficients in $\mathbb{Z}$.

Any help?

Thanks.

Answer 1

A $G$-grading (or: $G$-gradation) in a graded object (a $G$-graded, $k$-algebra $R$ or a $G$-graded, left $R$-module, $_{R}M$ with $k$ a field or a commutative ring) is usually defined with respect to a group $G$ and it can be shown that for any group $G$, a $G$-grading is the same thing as a (co)action of $(kG)^{*}$ (where $(kG)^{*}$ is the dual hopf algebra of the group hopf algebra $kG$) on the respective object.

A consequence of the above, is that if $G$ is a finite abelian group, in which case $kG\cong (kG)^{*}$ as hopf algebras (i.e. self-dual), then each $G$-grading is equivalent to a specific action of $G$ on the respecting object. $\mathbb{N}$ is not a group and I guess this is why it is custom to speak about a $\mathbb{Z}$-graded object but not an $\mathbb{N}$-graded object.

More generally, gradations can also be defined with respect to some monoid or semi-group, which means that it is legitimate to view your polynomial ring either as $\mathbb{Z}$-graded or $\mathbb{N}$-graded. So, the graded version of Nakayama's Lemma applies. However, in the literature it is the group-grading case which is mostly explored and which presents a wealth of behaviours (in the sense that it can provide deep insights into the structure of the graded ring, algebra or module).

Answer 2

You're totally right that it is $\mathbb{N}$-graded, and that you can use the graded Nakayama lemma over this ring. After all, the degree of any polynomial is nonnegative, regardless of what coefficients you're using. Either your professor made a mistake or you misunderstood what they were trying to say.