Primary ring

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 $$b^n = 0$$.

Stronger properties

 * Field
 * Integral domain