Prerequisites and Complements in Commutative Algebra

Abstract

Proposition 2.1. Let A be a subring of a ring B, and let x ? B. The following 5 assertions are equivalent: The element x is integral over A, i.e., there exists a monic polynomial $P(t) = t^n + a_1 t^{n-1} + ... + a_{n-1}t + a_n \in A [ t ]$ such that P(x) = 0. The subring A[x] of B generated by A and x is a finitely generated A–module. There exists a subring A′ of B, containing A[x], which is a finitely generated A–module. 12 The proof is classical and is left to the reader.