# a “strange” depth inequality

The following question arises in the context of the proof of Proposition 3.3.18 in Bruns and Herzog, Cohen-Macaulay Rings.

Let $R$ be a CM ring, not necessarily local, and suppose that $R$ admits a canonical module $\omega_R$, i.e. for every maximal ideal $m$ of $R$ we have that $(\omega_R)_m$ is the canonical module of $R_m$. Suppose in addition that

(a) $\omega_R$ has rank equal to $1$

(b) $\omega_R$ is isomorphic to a proper ideal of $R$ (call it also $\omega_R$)

(c) $\dim R >0$.

Let $p$ be a prime ideal of $R$ that contains $\omega_R$. Then Bruns and Herzog write in their proof that

$\operatorname{depth} \frac{R_p}{\omega_R R_p} \ge \operatorname{depth} R_p -1$.

Where does this inequality come from? I suppose B&H use the fact that $\omega_R R_p$ is a maximal CM $R_p$-module, but i still don't see how the inequality follows.