Let $R$ be a Noetherian ring (not necessarily local) and $I$ be the proper ideal of $R$, and $M$ be a finitely generated module such that $grade(I, M) = n > 1$, and $I$ has a generating set $\{x_1,...,x_n\}$. I want to show that least one of $x_i$ is a non-zero divisor on $M$.
By corollary 1.6.19 (Cohen-Macaulay Herzog) this statement is true when $R$ is local. I guess this is not true when $R$ is not local.
Can anyone help me find a counterexample?