Let $R$ be a commutative Noetherian domain (is not a field) with $\operatorname{Jac}(R)=(0)$. Why the set of maximal ideal of $R$ is infinite?
2026-03-28 10:24:16.1774693456
Why the set of maximal ideal of a domain with jacobson radical zero is infinite?
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If there were only finitely many maximal ideals $m_1,\ldots,m_n$ then pick non-zero $x_i\in m_i$ and you will find $\prod x_i\in Jac(R)=\bigcap m_i$. But certainly in a domain this product is not zero, so we have found a non-zero element in the Jacobson radical.