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  "body": "\n## Summary\n\nLet $M$ be a module over a commutative ring $R$.\nIts torsion submodule is the set of elements of $M$ that are killed by some non-zero-divisor of $R$.\nIt is a [[submodule|05WDSN597C]] of $M$, and it measures how far $M$ is from being torsion-free.\nFor $R = \\Z$ it is the [[torsion subgroup|0DBWXQRXQJ]] of an [[abelian group|082GHNPJ53]].\nOver a [[principal ideal domain|0CMD0BCH8T]], a finitely generated module splits as its torsion submodule plus a free module.\n\n## Explanations\n\n### L1 human\n\nThink of the elements of a module that \"vanish after scaling\" by a harmless scalar.\nA harmless scalar is one that is not a zero divisor, such as a nonzero integer.\nThese elements form the torsion part, and dividing them out leaves a torsion-free module.\nFor example, in the abelian group $\\Z \\oplus \\Z/6\\Z$, the element $(0, 1)$ is torsion because $6 \\cdot (0, 1) = 0$.\nThe element $(1, 0)$ is not torsion, because $n \\cdot (1, 0) = (n, 0) \\ne 0$ for every $n \\ne 0$.\n",
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