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

From Commalg

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 ringsVIEW 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 finitely many irreducible closed subsetsView other properties of commutative unital rings determined by the spectrum

## Contents

## 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).

## Relation with other properties

### Stronger properties

### Weaker properties

## 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).