Theorem

Krull dimension of an infinite product of zero dimensional rings

If $R_\alpha$ is a family of commutative rings with Krull dimension $0$, and $R=\prod_\alpha R_\alpha$, then the following are equivalent: 1) $R$ has Krull dimension $0$; 2) $R$ has finite Krull dimension; 3) $J(R)=N(R)$; 4) $N(R)=\prod_\alpha N(R_\alpha)$.

Reference(s)

R. W. Gilmer and W. Heinzer. Products of commutative rings and zero-dimensionality. (1992) @ Proposition 3.4 p 668