Classical quotient ring of a commutative reduced ring

If $R$ is a commutative reduced ring, the following are equivalent: 1) the classical quotient ring of $R$ is von Neumann regular; 2) The spectrum of minimal ideals of $R$ is compact in the Zariski topology, and if a finitely generated ideal is contained in a union of minimal primes, it is contained in one of them.


  • E. Matlis and others. The minimal prime spectrum of a reduced ring. (1983) @ Proposition 1.15 p 362
  • Y. Quentel. Sur la compacit{\'e} du spectre minimal d’un anneau. (1971) @ Proposition 9