A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element.

Question

I expect that the following result is true, but i can't prove it.

A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element.

I need some help to prove this.

Answer 1

Pick a generator $p$ of minimum degree $n$ for your ideal. Write $$p = p_n + p_{n-1} + \cdots + p_0$$ where the $p_d$ are homogeneous of degree $d$. Since $(p)$ is homogeneous and $p \in (p)$, we have $p_n \in p$. It follows that $p - p_n \in (p)$; this is a polynomial of degree less than $n$.

On the other hand, every element of your ideal is of the form $p f$ for some polynomial $f \in K[x_1, \ldots, x_n]$. In particular, every nonzero element of your ideal has degree at least $n$.

Therefore $p - p_n = 0$, so $p = p_n$, i.e. $p$ is homogeneous.