Ring where every prime contains a minimal prime

Symbol-free definition
A ring where every prime contains a minimal prime is a commutative unital ring satisfying the following: every prime ideal of the ring contains a minimal prime ideal.

Stronger properties

 * Integral domain: There's a unique minimal prime in this case: the zero ideal.
 * Noetherian ring:
 * Finite-dimensional ring
 * Ring where every prime contains a minimal prime and having finitely many minimal primes

Facts

 * In such a ring, the nilradical equals the intersection of all the minimal primes
 * A ring where every prime contains a minimal prime satisfies many of the nice properties that Noetherian rings do. For instance, in a reduced Noetherian ring, any zero divisor is in a minimal prime. It turns out that the assumption of Noetherianness can be weakened to the assumption that every prime contains a minimal prime