Transcendental element over a field The following is a lemma in a note of graded ring, however, I do not know how to prove it. Please help me. Thanks.
Let $R$ be a commutative reduced graded ring where $R_{0}$ is a field and let $u\in R_n\setminus \lbrace0\rbrace$. Then $u$ is transcendental over $R_{0}$.

Suppose there exists an integer $m$ and $a_{m - 1}, \ldots, a_0 \in R_0$ such that
$$
u^m + a_{m - 1}u^{m - 1} + \cdots + a_0 = 0.
$$
Now, $u^m$ is a non-zero [$R$ is reduced] element of degree $nm$. The above equation gives the additive inverse for $u^m$, which must also be a non-zero element of $R_{nm}$. But given that $R = \bigoplus R_d$, is this possible? What is the degree of $a_iu^i$?
