Irreducible element

Definition

Let $R$ be an integral domain. An element $p \in R$ is irreducible if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a unit.
3. If $p = a b$ with $a, b \in R$, then $a$ is a unit or $b$ is a unit.

Remarks

The irreducible elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ prime. In $\mathbb{Z}[i]$, the number $3$ is irreducible, but $5 = (2 + i)(2 - i)$ is not. Every prime element is irreducible. The converse can fail. For polynomials over a field, the irreducible elements are the irreducible polynomials.

Explanations

L1 human

An irreducible element cannot be split into smaller factors. The only factorizations are trivial ones, with a unit as one factor, like $7 = (-1)(-7)$. In the integers, the irreducible elements are the primes and their negatives.