Definition 10.52.1. Let $R$ be a ring. For any $R$-module $M$ we define the *length* of $M$ over $R$ by the formula

\[ \text{length}_ R(M) = \sup \{ n \mid \exists \ 0 = M_0 \subset M_1 \subset \ldots \subset M_ n = M, \text{ }M_ i \not= M_{i + 1} \} . \]

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