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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] with identity $1$.\nA subset $S \\subseteq R$ is a *multiplicative set* (or *multiplicatively closed set*) if:\n\n1. $1 \\in S$;\n2. $st \\in S$ for all $s, t \\in S$.\n\n## Remarks\n\nExample: let $\\mathfrak p$ be a [[prime ideal|00MHHZ9SNE]] of $R$ and $S = R \\setminus \\mathfrak p$.\nThen $1 \\in S$ because $\\mathfrak p$ is a proper ideal.\nIf $s, t \\notin \\mathfrak p$, then $st \\notin \\mathfrak p$ by the definition of a prime ideal, so $st \\in S$.\n\nOther examples are the powers $\\set{1, f, f^2, \\ldots}$ of an element $f \\in R$, the [[units|0XVEZDE9J8]] of $R$, and the set of elements of $R$ that are not [[zero divisors|0NNRDW10RZ]].\n\nNon-example: in $\\Z$, the set $S = R \\setminus 6\\Z$ is not multiplicative.\nIt contains $2$ and $3$ but not $2 \\cdot 3 = 6$.\nThe ideal $6\\Z$ is not prime.\n\nSome authors also require $0 \\notin S$.\nIf $0 \\in S$, the [[localization|4C1AE84N5N]] $S^{-1}R$ is the zero ring.\n",
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