How to understand $\mathbb{Z}^n$-graded ring?

Question

I am reading Ringel's note and I encountered a question I have never met. The question is what's the meaning of a $\mathbb{Z}^n$-graded ring？ This is from the following:

"Note that the rings $U_q (n_ +(\Delta))$ and is $\mathbb{Z}^n$-graded, where we assign to $E_i$ the degree $e_i$. Given $d \in \mathbb{Z}^n$, we denote by $U_q (n_ +(\Delta))_d$ the set of homogeneous elements of degree $d$." Which is from Ringel'note:

Note picture

Where $A^{'}$ is $\mathbb{Q}(v)$.

Answer 1

I assume you know what a graded ring is. This definition can be generalized to a $G$-graded ring for any monoid $G$, by replacing the role of the natural numbers or the integers by the monoid $G$. See here.

In this case, the grading is by the monoid $G = (\mathbb{Z}^n,+)$.

Answer 2

If $R$ is a ring and $G$ is any group, you can say that $R$ is $G$-graded if you have a direct sum decomposition $R = \bigoplus R_g$ as an additive group, such that $R_g\cdot R_h\subset R_{gh}$ for any $g,h\in G$. In particular it makes sense for $G=\mathbb{Z}^n$.