Prime ideal

Let $R$ be a commutative 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 $b\in P$.

Equivalently, $P$ is prime iff the quotient ring $R/P$ is an integral domain . 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 $(6)$.