The maximum condition on annihilators for polynomial rings

For each positive integer $n$, we construct a commutative ring ${\mathcal{R}}$ such that the polynomial ring ${\mathcal{R}}[x_{1},\ldots ,x_{n}]$ satisfies the maximum condition on annihilators and ${\mathcal{R}}[x_{1},\ldots ,x_{n+1}]$ does not. In particular, there exists a commutative Kerr ring ${\mathcal{R}}$ such that ${\mathcal{R}}[x]$ is not Kerr. This answers in the negative a question of Faith's.