Primary ring

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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

Definition

Symbol-free definition

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 R is termed a primary ring is whenever ab = 0 in R, then either a = 0 or there exists a n such that b^n = 0.

Relation with other properties

Stronger properties