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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a right $R$-[[module|0NZ8JA25WK]].\nThe module $M$ is *uniserial* if its set of [[submodules|05WDSN597C]] is [[totally ordered|0FRXBVVR03]] by inclusion.\nThat is, for any two submodules $N$ and $L$ of $M$, either $N \\subseteq L$ or $L \\subseteq N$.\nLeft uniserial modules are defined in the same way.\n\n## Remarks\n\nExample: for a prime $p$ and $n \\ge 1$, the $\\Z$-module $\\Z/p^n\\Z$ is uniserial.\nIts submodules are the subgroups $p^i\\Z/p^n\\Z$ for $0 \\le i \\le n$, and $p^i\\Z/p^n\\Z \\supseteq p^j\\Z/p^n\\Z$ when $i \\le j$.\nIn the same way, $k[x]/(x^n)$ is uniserial as a module over itself, for a field $k$.\n\nNon-example: the $\\Z$-module $\\Z$ is not uniserial.\nIts submodules $2\\Z$ and $3\\Z$ are not comparable, since $2 \\notin 3\\Z$ and $3 \\notin 2\\Z$.\n\nA nonzero uniserial module is indecomposable, since two nonzero submodules always meet in the smaller one, which is nonzero.\nA uniserial module of finite length has exactly one composition series.\n",
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