Is the product of Jacobson rings a Jacobson ring?

Question

I asked this question on Mathematics Stackexchange, but got no answer.

Is the product $$ \prod_{i\in I}A_i $$ of a family $(A_i)_{i\in I}$ of Jacobson rings a Jacobson ring?

(Here "ring" means "commutative ring with one".)

Answer 1

The answer is no in general.

Take $R = \prod_{n \in \mathbb{N}_{> 0}} \mathbb{Z}/2^n\mathbb{Z}$. Then the Jacobson radical of $R$ is $\prod_{n \in \mathbb{N}_{> 0}} 2\mathbb{Z}/2^n\mathbb{Z}$, and it contains a non-nilpotent element, namely $(2 + 2^n \mathbb{Z})_n$. Therefore the Jacobson radical of $R$ doesn't coincide with the nilradical of $R$.

All credits go to Georges Elencwajg, see this MO post.

Side note. Let $J(R)$ denote the Jacobson radical of a commutative unital ring $R$. Knowing that $x \in J(R)$ holds if and only if $1 + rx \in R^{\times}$ holds for every $r \in R$, the identity $J(\prod_{i \in I} A_i) = \prod_{i \in I} J(A_i)$ is immediate.