Hello!

This question may seem obvious but I cant seem to figure it out.

In the definition of an irreducible element (when developing the theory for a UFD foranygeneral Integral domain), it is required that we exclude all units from being irreducible. Why is this? In other words, what would happen if we did not exclude irreducibles from being units? Would this make every non zero element of the Integral domain uniquely factorizable into irreducible? I cant seem to trace the exact connection between preventing units from being irreducible and the unique factorization thereof...