integrality is transitive

Let CBA be rings. If B is integral over C and A is integral over B, then A is integral over C.

Proof. Choose uA. Then un+b1un-1++bn=0,biB. Thus C[b1,,bn,u] is integral and thus module-finite over C[b1,,bn]. Each bi is integral over C, so C[b1,,bn] is integral hence module-finite over C. Thus C[b1,,bn,u] is module-finite, hence integral, over C, so u is integral over C.

Title integrality is transitive
Canonical name IntegralityIsTransitive
Date of creation 2013-03-22 17:01:25
Last modified on 2013-03-22 17:01:25
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 13B21