Actually I got a confusion when I am learning the proof the theorem "An integral domain $R$ with ACCP is FD(Factorization Domain)."
The begining of the proof of the theorem goes as follows-
Suppose on contrary, $D$, which is an ID with ACCP, is not FD. Then $\exists$ a non-zero, non-unit element $a\in R$ such that a does not have factorization. Then $a$ is not irreducible. ...(and so on)
My question is why the non-zero, non-unit element $a \in R$, which does not have factorization, is not irreducible?
Can anybody clear my confusions?
What is the prime factorization of a prime number?