TITLE: Is $\mathbb{Z}[x,y,z]$ an $\mathbb{N}$-graded ring?
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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.