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  "body": "\n## Definition\n\nLet $k$ be a [[field|0QA8CHH8MG]], let $X$ be a finite set, and let $X^*$ be the [[free monoid|CV013KHVJG]] of words in $X$.\nLet $k\\langle X \\rangle$ be the [[free algebra|2P3Y2FRCSZ]] on $X$, whose basis is $X^*$.\nFix an *admissible order* $\u003c$ on $X^*$: a well-order such that $u \u003c v$ implies $p u q \u003c p v q$ for all words $p, q$.\nFor a nonzero $f \\in k\\langle X \\rangle$, its *leading word* $\\operatorname{LW}(f)$ is the largest word that occurs in $f$ with a nonzero coefficient.\nLet $I$ be a [[two-sided ideal|0DDA6CY7GJ]] of $k\\langle X \\rangle$.\nA subset $G \\subseteq I$ is a *(noncommutative) Gröbner basis* of $I$ if for every nonzero $f \\in I$ there are $g \\in G$ and words $p, q$ with\n$$\\operatorname{LW}(f) = p \\operatorname{LW}(g)\\, q.$$\n\n## Remarks\n\nA Gröbner basis $G$ of $I$ generates $I$ as a two-sided ideal.\nThe words that contain no $\\operatorname{LW}(g)$, $g \\in G$, as a subword are the *normal words*, and their images form a $k$-basis of $k\\langle X \\rangle / I$.\nExample: $X = \\set{x, y}$ with the degree-lexicographic order, $x \u003c y$, and $I$ generated by $g = yx - xy$, so $\\operatorname{LW}(g) = yx$.\nThe word $yx$ has no overlap with itself, so by the diamond lemma $\\set{g}$ is a Gröbner basis; the normal words are $x^a y^b$, and $k\\langle x, y \\rangle / I \\cong k[x, y]$.\nNon-example: with the same order, let $I$ be generated by $g = xx - y$, so $\\operatorname{LW}(g) = xx$.\nThe element $gx - xg = xy - yx$ lies in $I$, and its leading word is $yx$, which does not contain $xx$.\nSo $\\set{g}$ is not a Gröbner basis of $I$.\n",
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