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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nAn element $m \\in M$ is a *torsion element* if $r m = 0$ for some $r \\in R$ that is not a [[zero divisor|0NNRDW10RZ]] of $R$.\nThe *torsion submodule* of $M$ is the set\n$$T(M) = \\set{m \\in M : r m = 0 \\text{ for some non-zero-divisor } r \\in R}.$$\nThe module $M$ is *torsion-free* if $T(M) = 0$, and it is a *torsion module* if $T(M) = M$.\n\n## Remarks\n\nThe set $T(M)$ is a [[submodule|05WDSN597C]]: if $r m = 0$ and $s n = 0$ with $r, s$ non-zero-divisors, then $rs$ is a non-zero-divisor and $rs(m + n) = 0$, and $r(am) = a(rm) = 0$ for $a \\in R$.\nExample: for $R = \\Z$ and $M = \\Z \\oplus \\Z/6\\Z$, one has $T(M) = 0 \\oplus \\Z/6\\Z$, since $6 (k, c) = (6k, 0)$ is zero exactly when $k = 0$.\nNon-example: the element $(1, 0)$ of this $M$ is not torsion, because $n (1, 0) = (n, 0) \\ne 0$ for every nonzero integer $n$.\nWhen $R$ is an [[integral domain|0QVD7WBFGN]], the non-zero-divisors are the nonzero elements, and $M / T(M)$ is torsion-free.\nFor $R = \\Z$ the torsion submodule is the [[torsion subgroup|0DBWXQRXQJ]].\n",
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