Irreducible element

In ring theory a element $r$ of a ring $R$ is said to be irreducible if:

$r$ is not a unit.
$r$ cannot be written as the product of two non-units in $R$ , that is if $r = ab$ for some $a,b\in R$ then either $a$ or $b$ is a unit in $R$ .

This is analogous to the definition of prime numbers in the integers and indeed in the ring $\mathbb Z$ the irreducible elements are precisely the primes numbers and their negatives.

In a principal ideal domain it is easy to see that the ideal $(m)$ is maximal iff $m$ is irreducible. Indeed, we have $(m)\subseteq (a)$ iff $m|a$ so if $m$ is irreducible then $(m)\subseteq (a) \Rightarrow (a)=(m)$ or $(a)=(1)$ (since $a|m$ , either $a$ is a unit (so $(a)=(1)$ ) or $a$ is $m$ times a unit (so $(a)=(m)$ )). Conversely if $(m)$ is maximal then if $m=ab$ we have $a|m$ so $(m)\subseteq (a)$ hence either $(a)=(1)$ or $(a)=(m)$ . In the first case $a$ is a unit and in the second case $a=pu$ , where $u$ is a unit, and hence $b=u^{-1}$ , a unit. So in either case $m$ is irreducible.

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Irreducible element