Ring where every prime contains a minimal prime

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

This property of commutative unital rings is completely determined by the spectrum, viewed as an abstract topological space. The corresponding property of topological spaces is: union of irreducible closed subsets

View other properties of commutative unital rings determined by the spectrum

Definition

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.

Relation with other properties

Stronger properties

• Integral domain: There's a unique minimal prime in this case: the zero ideal.
• Noetherian ring: For full proof, refer: Noetheria ring has finitely many minimal primes and every prime contains a minimal prime
• 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