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  "body": "\n## Summary\n\nAn algebra in a class of algebras is injective when every homomorphism into it from a subalgebra extends to the whole algebra.\nIt is the universal-algebra form of an injective object, with embeddings in the role of monomorphisms.\nIn the variety of abelian groups the injective algebras are the divisible groups, and in modules they are the [[injective modules|85H7PRHDPZ]].\nInjective algebras and injective hulls are tools for embedding algebras into well-behaved ones.\n\n## Explanations\n\n### L1 human\n\nAn injective algebra is a target that never blocks an extension.\nIf a map is defined on part of a bigger algebra, it can always be extended to all of it.\nFor example, any homomorphism from the subgroup $\\Z$ of the rational numbers into $\\Q$ extends to a homomorphism from $\\Q$ to $\\Q$: $n \\mapsto nq$ extends to $x \\mapsto xq$.\nThe group $\\Z$ itself is not injective: the identity map $\\Z \\to \\Z$ has no extension to $\\Q$.\n",
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