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  "body": "\n## Summary\n\nA Noetherian module is a module in which every increasing chain of submodules stops after finitely many steps.\nEquivalently, every submodule is finitely generated.\nA ring is a [[Noetherian ring|0670T59HGE]] exactly when it is a Noetherian module over itself.\nThis finiteness condition makes induction arguments work and underlies most of commutative algebra.\n\n## Explanations\n\n### L1 human\n\nThink of building bigger and bigger submodules, one step at a time.\nIn a Noetherian module you cannot keep growing forever: the chain must become constant.\nFor example, in $\\Z$ seen as a module over itself, the chain $(12) \\subset (6) \\subset (3) \\subset \\Z$ cannot be made infinitely long upward.\nBy contrast, in $\\Q$ seen as a $\\Z$-module, the chain $\\Z \\subset \\frac{1}{2}\\Z \\subset \\frac{1}{4}\\Z \\subset \\cdots$ goes on forever.\n",
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