Let $R$ be a graded ring $R=\bigoplus^{\infty}_{i=-\infty} R_{i}$. Let $f=\sum f_{n}$ be a unit element in $R$. Suppose $f_{0}$ is not contained in any mininal (homogeneous) prime ideal of $R$, then we must have $f_{0}$ is a unit in $R_{0}$ and all other $f_{n}$ are nilpotent elements.
I know when $R$ is a non-negatively graded ring, we can inductively get $f_{n \geq 1}$ is nilpotent with the method from Atiyah's book Exercise 1.2. But when $R$ is a $\mathbb{Z}$-graded ring, I don't know how to solve it. Besides, I notice the constraint condition on $f_{0}$ is necessary, because $k[x,\frac{1}{x}]$ is a good example where $x$ is a unit.
Could anyone give me some hint or tell me how to do? Thanks!
