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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $M$ be a left [[module|09VDS2W1E8]] over $R$.\nA chain of [[submodules|0RGZ10DFPG]] $M_0 \\subsetneq M_1 \\subsetneq \\cdots \\subsetneq M_n$ of $M$ has length $n$.\nThe *length* $\\ell_R(M)$ of $M$ is the supremum of the lengths of all such chains, an element of $\\set{0, 1, 2, \\dots} \\cup \\set{\\infty}$.\n\n## Remarks\n\nExample: for a finite-dimensional vector space $V$ over a field $k$, $\\ell_k(V) = \\dim_k V$.\nIndeed, a strict chain of subspaces raises the dimension at each step, and a chain built from a basis has $\\dim_k V$ steps.\nNon-example of finite length: $\\ell_\\Z(\\Z) = \\infty$, because $2^n\\Z \\subsetneq 2^{n-1}\\Z \\subsetneq \\cdots \\subsetneq \\Z$ has length $n$ for every $n$.\nA composition series of $M$ is a chain $0 = M_0 \\subsetneq \\cdots \\subsetneq M_n = M$ whose factors $M_i/M_{i-1}$ are simple modules.\nA module $M$ has finite length if and only if it has a composition series, and then $\\ell_R(M) = n$ by the [[Jordan–Hölder theorem for modules|M1ABX18FC4]].\nFor a short exact sequence $0 \\to M' \\to M \\to M'' \\to 0$, one has $\\ell_R(M) = \\ell_R(M') + \\ell_R(M'')$.\nThe group version of a composition series is the [[composition series of a group|0NB8VPB93V]].\n\n## Explanations\n\n### L1 human\n\nThe length of a module counts the steps in the longest strict chain of submodules.\nIt is the module version of dimension.\n",
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