Prime ideal

Definition

Let $R$ be a commutative ring. An ideal $P$ of $R$ is prime if it has these two properties.

1. $P \neq R$.
2. For all $a, b \in R$, if $a b \in P$, then $a \in P$ or $b \in P$.

Remarks

In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\mathbb{Z}$ is also prime. The zero ideal of $R$ is prime if and only if $R$ is an integral domain. An ideal $P$ is prime if and only if $R/P$ is an integral domain. Every maximal ideal is prime.

Explanations

L1 human

A prime number $p$ has this property: if $p$ divides $a b$, then $p$ divides $a$ or $b$. A prime ideal has the same property, with "divides" replaced by "contains". The ideal $(6)$ is not prime: it contains $2 \cdot 3$ but neither $2$ nor $3$.