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  "body": "\n## Summary\n\nThe free algebra $k\\langle X \\rangle$ on a set $X$ over a [[commutative ring|0310REJJY8]] $k$ is the algebra of polynomials in noncommuting variables from $X$.\nIt is the free object among [[associative algebras|3CTW1P4HBP]] over $k$: any map from $X$ into an algebra extends uniquely to an algebra homomorphism.\nSo every algebra generated by a set $X$ is a quotient of $k\\langle X \\rangle$, and algebras are often given by generators and relations in this way.\nIt is the noncommutative analogue of the [[polynomial ring|0SDC934R52]] and is isomorphic to the [[tensor algebra|P4J5JZDW0B]] of the free module with basis $X$.\n\n## Explanations\n\n### L1 human\n\nTake letters $x, y$ and form all words in them, such as $xy$, $yx$, $xxy$.\nUnlike in ordinary polynomials, $xy$ and $yx$ are different words.\nThe free algebra consists of finite linear combinations of words, like $3xy - 2yx + 5$, multiplied by joining words end to end.\nEvery other algebra with two chosen elements is obtained from this one by imposing rules, for example $xy = yx$ gives back the polynomial ring $k[x, y]$.\n",
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