Definition 10.68.1. Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements $f_1, \ldots , f_ r$ of $R$ is called an *$M$-regular sequence* if the following conditions hold:

$f_ i$ is a nonzerodivisor on $M/(f_1, \ldots , f_{i - 1})M$ for each $i = 1, \ldots , r$, and

the module $M/(f_1, \ldots , f_ r)M$ is not zero.

If $I$ is an ideal of $R$ and $f_1, \ldots , f_ r \in I$ then we call $f_1, \ldots , f_ r$ an *$M$-regular sequence in $I$*. If $M = R$, we call $f_1, \ldots , f_ r$ simply a *regular sequence* (in $I$).

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