are units considered irreducible

moont14263

Definition: Let R be an integral domain. A nonzero, nonunit element r in R is said to be irreducible if whenever r=ab, then a is a unit or b is a unit.

My question is are units considered irreducible.

This how I understand it,
Let v in R be a unit such that v=ab ==> 1=ab(v^-1) ==> 1=a[b(v^1)] ==> a is unit.
So according to this, v is irreducible.
Am I right? Help!!!!
Thanks.

zhentil

Huh? Is the question about which word was being defined in the definition?

mathwonk

no. units are not irreducibles by definition.