Definition. prime ideal [math-0012]

Let $A$ be a ring and $I$ be an ideal, the followings are equivalent conditions to say that $I$ is prime

1. $I$ is prime if $ab \in I$ than $a \in I$ or $b \in I$ for all $a,b \in A$
2. $I$ is prime if $A / I$ is an integral domain

Proof

Backward

Let $A / I$ be an integral domain, that's say if $x, y \in A / I$ and $xy = 0$ , then $x = 0$ or $y = 0$ . Let $(a + I)(b + I)$ be the zero element of $I$ (i.e. $(0 \in A) + I$ ), then $ab + I = I$ . Hence $a + I = I$ or $b + I = I$ , implies $a \in I$ or $b \in I$ .

Forward

Let $I$ be a prime ideal, let

then $ab \in I$ and therefore, $a \in I$ or $b \in I$ . Hence $a + I$ or $b + I$ is the zero coset in $A / I$ .