Prime ideal - Knowlpedia

Let $R$ be a commutative unital ring. A prime ideal is a proper ideal $P\subsetneq R$ such that whenever $ab\in P$ (with $a,b\in R$ ), then $a\in P$ or $b\in P$ .

Equivalent characterizations

Equivalently, $P$ is prime iff the quotient ring $R/P$ is an integral domain.

Remarks

Prime ideals generalize prime numbers and underpin dimension theory, localization, and algebraic geometry.

Examples

In $\mathbb Z$ , the ideal $(p)$ is prime iff $p$ is a prime integer.
In $k[x,y]$ , the ideal $(x)$ is prime, and so is $(x,y)$ .
In $\mathbb Z$ , the ideal $(6)$ is not prime since $2\cdot 3\in (6)$ but neither $2$ nor $3$ lies in $(6)$ .