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Let $\lbrace R_i \rbrace_{i\in I}$ be a family of rings with unity, where $J_i$ is the Jacobson radical of $R_i$. Can we conclude that $$ J\left(\prod_{i \in I} R_i \right) = \prod_{i \in I} J(R_i) = \prod_{i \in I} J_i \ ? $$

When $I$ is a finite index set, the above equation holds. What about an infinite index set?

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1 Answer 1

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This is obvious via the following characterization:

$x\in J(R)$ if and only if for every $r\in R$ the element $1-xr$ is a unit of $R$.

Applying this to an element $\vec{x}\in \prod R_i$, it is not hard to see this holds for $\vec{x}$ iff it holds for all its components.

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