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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]].\nA *monomial* of $S$ is a product $x^a = x_1^{a_1} \\cdots x_n^{a_n}$ with $a = (a_1, \\ldots, a_n) \\in \\N^n$.\nAn [[ideal|096Z9MP2ME]] $I$ of $S$ is a *monomial ideal* if it is generated by monomials.\n\n## Remarks\n\nAn equivalent form: $I$ is a monomial ideal if and only if, for every $f \\in I$, each monomial that occurs in $f$ with a nonzero coefficient lies in $I$.\nA monomial $x^b$ lies in the monomial ideal generated by monomials $x^{a}$, $a \\in A$, if and only if $b - a \\in \\N^n$ for some $a \\in A$.\nBy Dickson's lemma, every monomial ideal is generated by finitely many monomials, and it has a unique minimal set of monomial generators.\n\nExample: $I = (x^2, xy)$ in $k[x, y]$ is a monomial ideal by definition.\nThe polynomial $x^3 + x y^5$ lies in $I$, and indeed both of its monomials are multiples of $x^2$ or of $xy$.\n\nNon-example: $I = (x + y)$ in $k[x, y]$ is not a monomial ideal.\nIt contains $x + y$, but it does not contain the monomial $x$, since every nonzero element of $I$ is a multiple of $x + y$.\n",
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