I just studied UFD definition, and I figured out 'factorisable' and 'uniqueness' are independent.
Classical example $Z[\sqrt{-5}]$ is ring that every element and be factorisable but not uniquely.
I thought example $Z^N$ that not every element can be factorisable but every factorisable element can be uniquely factorized.
From those examples, I thought those questions:
(1) What is example of ring such as $Z^N$ without infinite tuple things?
(2) With only property 'every factorisable element can be uniquely factorized' (let call that ring UFD'), does property of UFD such as "prime iff irreducible" and "any of two elements has GCD/LCM" hold? ($Z^N$ holds this property.)
(3) What is properties of ring of factorization domain without uniqueness? Above property are strongly depends on uniqueness of factorization rather than factorization.
PS. For question (2), I think this is false because $R^N$ for any UFD $R$ hold those properties. It is almost same as question (1), finding UFD' other than this naive construction.