Definition 10.72.1. Let $R$ be a ring, and $I \subset R$ an ideal. Let $M$ be a finite $R$-module. The $I$-depth of $M$, denoted $\text{depth}_ I(M)$, is defined as follows:

1. if $IM \not= M$, then $\text{depth}_ I(M)$ is the supremum in $\{ 0, 1, 2, \ldots , \infty \}$ of the lengths of $M$-regular sequences in $I$,

2. if $IM = M$ we set $\text{depth}_ I(M) = \infty$.

If $(R, \mathfrak m)$ is local we call $\text{depth}_{\mathfrak m}(M)$ simply the depth of $M$.

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