integrality is transitive

Let $C\\subset B\\subset A$ be rings. If $B$ is integral over $C$ and $A$ is integral over $B$ , then $A$ is integral over $C$ .

Proof. Choose $u\\in A$ . Then $u^{n}+b_{1}u^{n-1}+\\cdots+b_{n}=0,b_{i}\\in B$ . Thus $C[b_{1},\\ldots,b_{n},u]$ is integral and thus module-finite over $C[b_{1},\\ldots,b_{n}]$ . Each $b_{i}$ is integral over $C$ , so $C[b_{1},\\ldots,b_{n}]$ is integral hence module-finite over $C$ . Thus $C[b_{1},\\ldots,b_{n},u]$ is module-finite, hence integral, over $C$ , so $u$ is integral over $C$ .

单词	IntegralityIsTransitive
释义	integrality is transitive Let $C\\subset B\\subset A$ be rings. If $B$ is integral over $C$ and $A$ is integral over $B$ , then $A$ is integral over $C$ . Proof. Choose $u\\in A$ . Then $u^{n}+b_{1}u^{n-1}+\\cdots+b_{n}=0,b_{i}\\in B$ . Thus $C[b_{1},\\ldots,b_{n},u]$ is integral and thus module-finite over $C[b_{1},\\ldots,b_{n}]$ . Each $b_{i}$ is integral over $C$ , so $C[b_{1},\\ldots,b_{n}]$ is integral hence module-finite over $C$ . Thus $C[b_{1},\\ldots,b_{n},u]$ is module-finite, hence integral, over $C$ , so $u$ is integral over $C$ .
随便看	WeakHomotopyEquivalence weak homotopy equivalence WeakHopfAlgebra weak Hopf algebra WeakHopfCalgebra weak Hopf C-algebra WeaklyCompactCardinal weakly compact cardinal WeaklyCompactCardinalsAndTheTreeProperty weakly compact cardinals and the tree property WeaklyCountablyCompact weakly countably compact WeaklyHolomorphic weakly holomorphic WeakPrime weak prime WeakTopology weak- topology WeakTopologyOfTheSpaceOfRadonMeasures weak-* topology of the space of Radon measures WedderburnArtinTheorem Wedderburn-Artin theorem WedderburnEtheringtonNumber Wedderburn-Etherington number WedderburnsTheorem