Is an easy exercise to prove the following assert:
Let $B_0, \dots , B_n$ integral algebras over $A$. Then $\bigoplus_{i=0}^n B_i$ is integral over $A$.
Is this true if the sum is infinite? What about direct product?
I suppose the statement is false in this cases, but I've no counterexamples. Any idea?
Thank you.
You don't mean the direct sum, you mean the direct product $\prod_{i=0}^{n} B_i$ (SE/345501). The proof is straight forward for these finite products, and it also reveals what goes wrong in the infinite case: The degree of the integral equations may tend to infinity. So for example consider $\prod_{n=1}^{\infty} \mathbb{Z}[x]/(x^n-2)$. Clearly each $\mathbb{Z}[x]/(x^n-2)$ is integral over $\mathbb{Z}$, but the product is not, since the element $(x)_{n \geq 1}$ is not integral. For infinite "direct sums" we have no problems unless we face the problem that they don't have a unit, so that usual commutative ring theory breaks down (and that's why rings without unit are often called pseudo-rings or rngs).
