An irreducible element of a ring is a non-unit such that it cannot be written as a product of two non-units.
A UFD is a domain such that each non-unit $x\in R \backslash \{0\}$ can be written as a product of irreducible elements.
So if we take an irreducible element $r$ in a UFD, then it's not a unit and so admits a unique factorization into irreducible elements $r_1\dots r_n$ (which are also non-units). As it's irreducible it cannot be written as a product of two non-units so if $r = ab$ then $a$ or $b$ is a unit.
Does this mean that irreducible elements can be multiplied together to form units? And therefore non-units can be multiplied together to form units?
Best Answer
In a factorization, note that $n$ can be 1 (or zero!).
In general if $ab$ is a unit, then there exists $c$ so that $(ab)c=1$. Now, what can you say about $a$?