Primary ring: Difference between revisions

Latest revision as of 16:28, 12 May 2008

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

@@ Line 6: / Line 6: @@
 ===Symbol-free definition===
-A [[commutative unital ring]] is termed a '''primary ring''' if whenever the product of two elements in it is zero, either the first element is zero, or the second element is nilpotent.
+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 to a module|associated prime]].
 ===Definition with symbols===