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?