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  "body": "\n## Definition\n\nLet $A$ and $B$ be commutative rings and let $f \\colon A \\to B$ be a [[ring homomorphism|0GGE0Z0JQ1]].\nThe ring $B$ is an [[$A$-module|0NZ8JA25WK]] by $a \\cdot b = f(a) b$.\nThe homomorphism $f$ is *finite*, and $B$ is a *finite $A$-algebra*, if $B$ is a [[finitely generated|05QY1VEBYD]] $A$-module.\nThat is, there are $b_1, \\ldots, b_n \\in B$ such that every $b \\in B$ can be written as $b = f(a_1) b_1 + \\cdots + f(a_n) b_n$ with $a_i \\in A$.\n\n## Remarks\n\nExample: $\\Z \\to \\Z[\\sqrt{2}]$ is finite, generated as a module by $1$ and $\\sqrt{2}$.\nEvery surjective homomorphism $A \\to A/I$ is finite, generated by $1$.\n\nNon-example: $\\Z \\to \\Q$ is not finite.\nIf $q_1, \\ldots, q_n$ generated $\\Q$ as a $\\Z$-module, every rational number would have a denominator dividing the product of their denominators, which fails for $1/p$ with $p$ a large prime.\n\nA homomorphism is finite if and only if it is [[integral|B7X0KFY1S1]] and of [[finite type|AC5TFFQT9M]].\nA composition of finite homomorphisms is finite.\n",
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