I $A$ is a principal ideal domain and let N the set of non zero elements, that do not have an inverse and that cannot be decomposed in a product of irreductibles.
If $a \in N$
Let's create the set $I = aA$
I am trying to understand why a is not irreductible
(This is part of the proof of the fundamental theorem of arithmetics our professor gave us)
Is it just that if $a$ was irreductible then $a = a$ so it can be decomposed in a product of irreductibles?