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  "body": "\n## Definition\n\nLet $M$ be a commutative [[monoid|0N6R902FG1]], written additively.\nA *Grothendieck group* of $M$ is an [[abelian group|082GHNPJ53]] $K(M)$ with a monoid homomorphism $i \\colon M \\to K(M)$ with this universal property.\nFor every abelian group $A$ and monoid homomorphism $f \\colon M \\to A$, there is exactly one group homomorphism $g \\colon K(M) \\to A$ with $g \\circ i = f$.\n\n## Remarks\n\nIt exists and is unique up to unique isomorphism.\nOne construction: on $M \\times M$ put $(m, n) \\sim (m', n')$ if $m + n' + p = m' + n + p$ for some $p \\in M$.\nThe classes form an abelian group under componentwise addition, $i(m)$ is the class of $(m, 0)$, and the class of $(m, n)$ is $i(m) - i(n)$.\nExample: for $M = \\N$ under addition, the inclusion $\\N \\to \\Z$ is a Grothendieck group, since a homomorphism $f$ from $\\N$ extends uniquely by $g(k) = k f(1)$.\nNon-example: the inclusion $\\N \\to \\Q$ is not, because the identity map $\\N \\to \\N \\subseteq \\Z$ has no extension to a group homomorphism $\\Q \\to \\Z$.\nThe map $i$ is injective exactly when $M$ is cancellative; for $M = \\N \\cup \\set{\\infty}$ with $n + \\infty = \\infty$, one gets $K(M) = 0$.\n",
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