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  "body": "\n## Definition\n\nLet $k$ be a [[commutative ring|0310REJJY8]] with identity and let $X$ be a set.\nLet $X^*$ be the [[free monoid|CV013KHVJG]] on $X$, with product given by concatenation.\nIts elements are the finite words $x_1 x_2 \\cdots x_n$ with $x_i \\in X$, including the empty word $1$.\nThe *free algebra* $k\\langle X \\rangle$ is the free $k$-module with basis $X^*$, with the multiplication that extends concatenation of words $k$-bilinearly:\n$$\\Big(\\sum_{u} a_u \\, u\\Big)\\Big(\\sum_{v} b_v \\, v\\Big) = \\sum_{u, v} a_u b_v \\, uv.$$\nIt is an [[associative algebra|3CTW1P4HBP]] over $k$ with identity the empty word.\n\n## Remarks\n\nUniversal property: for every associative $k$-algebra $A$ with identity and every map $f \\colon X \\to A$, there is a unique homomorphism of $k$-algebras $\\bar f \\colon k\\langle X \\rangle \\to A$ with $\\bar f(x) = f(x)$ for $x \\in X$.\nIt sends a word $x_1 \\cdots x_n$ to $f(x_1) \\cdots f(x_n)$.\n\nExample: for $X = \\set{x}$, the words are the powers $x^n$, and $k\\langle x \\rangle$ is the [[polynomial ring|0SDC934R52]] $k[x]$.\n\nNon-example: for $X = \\set{x, y}$, the polynomial ring $k[x, y]$ is not free on $X$.\nThe map $x \\mapsto e_{12}$, $y \\mapsto e_{21}$ into $2 \\times 2$ matrices over $k$ has no extension to a homomorphism from $k[x, y]$, because $e_{12} e_{21} = e_{11} \\ne e_{22} = e_{21} e_{12}$ while $xy = yx$ in $k[x, y]$.\n\nThe free algebra is the [[monoid ring|1HM7ZYMPD7]] $k[X^*]$.\nIt is isomorphic to the [[tensor algebra|P4J5JZDW0B]] $T(V)$ of the free $k$-module $V$ with basis $X$, graded by word length.\nFor $\\abs{X} \\ge 2$ and $k$ a field, it is a domain that is not commutative.\n",
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