Integral extension is a finitely generated $R$-module?

Question

Let $R$ be a commutative ring. If $b_1,\ldots,b_n$ are elements of a ring $R'$ (commutative) which are integral over $R$ then $R[b_1,\ldots,b_n]$ is a f.g. $R$-module.

My question is: If $\{b_i\}_{i\in I}$ are elements integral over $R$, is then $R[\{b_i\}_{i\in I}]$ a f.g $R$-module?

Thanks for any counterexample.

Answer 1

The algebraic closure $\overline {\mathbf Q}\,$ of $\mathbf Q$ is not finitely generated over $\mathbf Q$ (as a vector space) since there exists irreducible polynomials of arbitrary degree.

Answer 2

You cannot adjoin an infinite number of elements and expect the module is still finitely generated. Take $R= \mathbb{Z}$ and $b_i = 2^{1/2^i}$ for example.