Definition 111.9.1. Let $A$ be a ring. Let $M$ be an $A$-module. The *length* of $M$ as an $R$-module is

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

In other words, the supremum of the lengths of chains of submodules.

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