Generators for a finitely generated graded ring

Question

Given a Noetherian graded ring (commutative and with 1) $A=\bigoplus_{n=0}^\infty A_n$, that's generated as an $A_0$-algebra by $x_1,\ldots, x_s\in A$. I am having difficulties seeing why there is no loss in generality by assuming that the $x_i$ are homogeneous. Could someone explain this to me?

Thanks in advance.

Answer 1

Write $x_i=\sum_{n\ge0}x_{in}$ with $x_{in}\in A_n$. Then $x_{in}$, $i=1,\dots,s$, $n\ge0$ is a homogeneous generating set for the $A_0$-algebra $A$.