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
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