Prime element

Definition

Let $R$ be a commutative ring. An element $p \in R$ is prime if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a unit.
3. For all $a, b \in R$, if $p \mid a b$, then $p \mid a$ or $p \mid b$.

Remarks

A nonzero $p$ is prime if and only if the principal ideal $(p)$ is a prime ideal. The prime elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ a prime number. In an integral domain, prime elements are irreducible.

Explanations

L1 human

Euclid's lemma says that a prime number dividing a product divides one of the factors. A prime element of a ring is an element with this property. In the integers, $6$ is not prime: $6$ divides $2 \cdot 3$, but not $2$ or $3$.