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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nThe module $M$ is *indecomposable* if:\n\n1. $M \\ne 0$;\n2. whenever $M = N_1 \\oplus N_2$ for [[submodules|0RGZ10DFPG]] $N_1, N_2$ of $M$, one has $N_1 = 0$ or $N_2 = 0$.\n\n## Remarks\n\nEquivalently, $M \\ne 0$ and the only [[idempotents|0X4JKZXJ86]] of the [[endomorphism ring|AWSBZWWMG9]] $\\End_R(M)$ are $0$ and $1$.\nExample: the $\\Z$-module $\\Z$ is indecomposable.\nThe check: two nonzero submodules $a\\Z$ and $b\\Z$ both contain $ab \\ne 0$, so their sum is never direct.\nExample: $\\Z/p^n\\Z$ is indecomposable for $p$ prime and $n \\ge 1$, since its subgroups form a chain.\nNon-example: $\\Z/6\\Z = \\set{0, 3} \\oplus \\set{0, 2, 4}$ fails clause 2.\nEvery simple module is indecomposable, but $\\Z/4\\Z$ is indecomposable and not simple.\n",
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