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  "body": "\n## Definition\n\nLet $A$ be a commutative ring and let $B$ be a commutative [[$A$-algebra|DPBJ04JMH8]] with structure map $f \\colon A \\to B$.\nThe algebra $B$ is *of finite type* over $A$, or *finitely generated* as an $A$-algebra, if it has a finite generating set.\nThat means there are $b_1, \\ldots, b_n \\in B$ such that every element of $B$ is $p(b_1, \\ldots, b_n)$ for a polynomial $p$ with coefficients in $f(A)$.\nEquivalently, there is a surjective $A$-algebra homomorphism from the [[polynomial ring|0SDC934R52]] $A[x_1, \\ldots, x_n]$ onto $B$ for some $n \\ge 0$.\n\n## Remarks\n\nExample: $k[x, x^{-1}]$ is of finite type over a field $k$.\nIt is generated by $x$ and $x^{-1}$, and it is isomorphic to $k[x, y]/(xy - 1)$.\n\nNon-example: $\\Q$ is not of finite type over $\\Z$.\nSuppose $\\Q = \\Z[q_1, \\ldots, q_n]$, and let $d$ be the product of the denominators of the $q_i$.\nThen every element of $\\Q$ would have a power of $d$ as a denominator, which fails for $1/p$ with $p$ a prime not dividing $d$.\n\nA [[finite|1NBACAEC2V]] algebra is of finite type, but $k[x]$ over $k$ shows that the converse fails.\nA finitely generated algebra over a Noetherian ring is Noetherian, by [[Hilbert's basis theorem|0YKA9J3XEE]].\n",
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