I overcame a theorem which says:
Let R be an Integral Domain. If every Irreducible element is Prime then it satisfies UFD 2. Converse (i.e, UFD 2 implies every Irreducible element is Prime) is true if it is UFD 1.
UFD 1 is basically the Existence of a Factorization of every element.
UFD 2 is the Uniqueness of the Factorization for every factorization of every element.
We know that being UFD satisfies both UFD 1 and UFD 2.
Now I was having trouble finding examples for the situation when UFD 2 satisfies but NOT UFD 1.
Can anyone help me with this? Thanks in advance. ( Please write in a bit details if possible)
My basic idea was to figure out examples for the case that UFD 2 does not imply that Every Irreducible element is Prime by avoiding UFD 1 ( if factorization exists then it has to be unique but it is not mandatory that every element will have factorization).