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  "body": "\n## Summary\n\nA ring homomorphism $f \\colon A \\to B$ is finite if $B$, viewed as an $A$-module through $f$, is [[finitely generated|05QY1VEBYD]].\nFinite homomorphisms are exactly the [[integral|B7X0KFY1S1]] homomorphisms that are also of [[finite type|AC5TFFQT9M]].\nThey are closed under composition and base change.\nIn algebraic geometry they correspond to finite morphisms of affine schemes, such as the projection of a curve onto a line in general position.\n\n## Explanations\n\n### L1 human\n\nA finite ring map makes the big ring a finite-dimensional object over the small ring, in the sense of modules.\nThe Gaussian integers $\\Z[i]$ are finite over $\\Z$: every element is $a \\cdot 1 + b \\cdot i$ with $a, b \\in \\Z$.\nThe polynomial ring $k[x]$ is not finite over the field $k$: the powers $1, x, x^2, \\ldots$ are linearly independent, so no finite set spans it.\nFiniteness is the ring version of a finite field extension.\n",
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