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:

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 $I$,

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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