I am searching for a commutative ring with zero Krull dimension which have infinitely many prime ideals.
Any help or suggestion would be appreciated!
2026-03-28 04:52:14.1774673534
Zero-dimensional ring with infinitely many maximal ideals
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The class of these rings can be totally classified as the class of rings having nil Jacobson radical $J(R)$ and having $R/J(R)$ a nonNoetherian von Neumann regular ring.
So an infinite product of fields works, or an infinite product of commutative VNR rings works. You could also take any (nonzero) module over such a ring and the idealization of the module would give you a version with nontrivial Jacobson radical.