A note on Krull dimension of skew polynomial rings

Let $A$ be a commutative Noetherian ring such that Krull dimension of $A$ is $\alpha$. Let $M$ be a finitely generated critical module over $A[x,\sigma]$, (where $\sigma$ is an automorphism of $A$) and Krull dimension of $M$ is $\alpha+1$. Then $M$ has a prime annihilator.