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I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, what about locali...
In a PID every non-zero,non-unit element can be written as product of irreducibles". This statement claims that a PID is an atomic domain, i.e., a domain where every non-zero, non unit admit a factorization into irreducibles (aka atoms, hence the name atomic). As you noted, being a UFD requires the uniqueness of such a factorization and this leads to another definition: an AP-domain is a ...
The fact that $A$ is a UFD implies that $A [X]$ is a UFD is very standard and can be found in any textbook on Algebra (for example, it is Proposition 2.9.5 in these notes by Robert Ash).
abstract algebra - Does $A$ a UFD imply that $A [T]$ is also a UFD ...
The point of a UFD is that any element can be rewritten as a product of irreducible factors, where any other product of irreducible factors is just a rearrangement of the exact same terms, is this correct?