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
The property of being an ideal for which the quotient ring has this property is: primary ideal
A commutative unital ring is termed a primary ring if it satisfies the following equivalent conditions:
- Whenever the product of two elements in it is zero, either the first element is zero, or the second element is nilpotent
- The zero ideal is a primary ideal
- The ring, as a module over itself, has a unique associated prime.
Definition with symbols
A commutative unital ring is termed a primary ring is whenever in , then either or there exists a such that .