If $R$ is a product ring whose factors are in a finite number and are all quotients of $\mathbb{Z}$ (that is, either $\mathbb{Z}$ or $\mathbb{Z}_n$'s ), is it a sufficient and necessary condition for the diagonal image of $\mathbb{Z}$ to be undefinable in it by a finite first-order formula, that there be at least two copies of $\mathbb{Z}$ among the factors?
2026-04-06 02:53:24.1775444004
Definibility of $\mathbb{Z}$ in product rings
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