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  "body": "\n## Definition\n\nFix a signature and let $\\mathcal{K}$ be a class of algebras of this signature.\nAn algebra $A \\in \\mathcal{K}$ is *injective* in $\\mathcal{K}$ if the following holds.\nFor all $B, C \\in \\mathcal{K}$, every [[embedding|SCRB74SWQN]] $g \\colon B \\to C$ and every [[homomorphism|KBXSZC6BV1]] $f \\colon B \\to A$, there is a homomorphism $h \\colon C \\to A$ with $h \\circ g = f$.\n\n## Remarks\n\nWhen $\\mathcal{K}$ is a variety, the embeddings are exactly the monomorphisms of the category of algebras of $\\mathcal{K}$.\nSo in a variety, $A$ is injective in this sense exactly when it is an injective object of that category.\n\nExample: $\\Q$ is injective in the variety of abelian groups.\nAn abelian group is injective there exactly when it is divisible, and $\\Q$ is divisible since $x = n \\cdot (x/n)$ for every $x \\in \\Q$ and $n \\ge 1$.\nIn the variety of modules over a ring, the injective algebras are the [[injective modules|85H7PRHDPZ]].\n\nNon-example: $\\Z$ is not injective in the variety of abelian groups.\nLet $g \\colon \\Z \\to \\Q$ be the inclusion and $f = \\id_{\\Z}$.\nA homomorphism $h \\colon \\Q \\to \\Z$ with $h(1) = 1$ would give $2 h(1/2) = 1$ in $\\Z$, which is impossible.\n\nIn many varieties, such as groups, only the trivial algebra is injective.\n",
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