From Theorem 13.7 ii) in the original edition (written in Japanese) of Matsumura's "Commutative Ring Theory," the following statement is got rid of:

Let $A = \bigoplus_{n \geq 0}A_n$ be a noetherian graded ring. If $P \subset A$ is a homogeneous prime ideal of height $r$, then there exists an ideal $I = (b_1, \ldots, b_r)$ generated by $r$ homogeneous elements $b_i$ such that $P$ is a minimal prime divisor of $I$.

The proof is written as follows:

The case $r = 0$ (hence $I = (0)$) is no problem. Assume $r > 0$. By Theorem 13.6, we can choose $a_1, \ldots, a_r \in P$ such that $P$ is a minimal prime divisor of $(a_1, \ldots, a_r)$. Put $J = (a_1, \ldots, a_r)$, and let $a_{ij}$ be the homogeneous term of $a_i$ of degree $j$. Since $JA_P$ is generated by $\{a_{ij}\}_{i, j}$, we can take a minimal basis from $\{a_{ij}\}$. Hence there exist $r$ homogeneuous elements $b_1, \ldots, b_r \in P$ with $JA_P = (b_1, \ldots, b_r)A_P$. Then $P$ is a minimal prime divisor of $(b_1, \ldots, b_r)A$.

Here is my question.

- Where are the mistakes in the proof above?
- What is a counterexample against the removed statement?