Ring where every prime contains a minimal prime and having finitely many minimal primes

Definition
A ring where every prime contains a minimal prime and having finitely many minimal primes is a commutative unital ring satisfying the following two conditions:


 * Every prime ideal of the ring contains a minimal prime ideal
 * There exist only finitely many minimal prime ideals

Equivalently, it is a ring whose spectrum has the following property: it can be expressed as a union of finitely many irreducible closed subsets (these irreducible closed subsets correspond to the minimal primes).

Stronger properties

 * Noetherian ring
 * Integral domain

Weaker properties

 * Ring where every prime contains a minimal prime

Facts

 * In such a ring, the nilradical equals the intersection, and hence, contains the product, of all the minimal primes (note that we need only finitely many minimal primes to make sense of their product)
 * Further, the intersection of the minimal primes is irredundant. That is because there are only finitely many of them, and no minimal prime contains another. Thus, for instance, this condition guarantees that every element in a minimal prime must be a zero divisor. (The result is typically stated for Noetherian rings as: in a Noetherian ring, every element in a minimal prime is a zero divisor).