For example, integers, gaussian integers, and polynomials all have unique factorizations. What are these rings (or this property) referred to as? Or is unique factorization a ubiquitous property that applies to all rings? If not, what is an example of a ring that doesn't have any meaningful "unique factorization" property?
I'm no expert at abstract algebra, so sorry if this is a silly question, or I'm using the term "ring" incorrectly (perhaps field or group is the more applicable term).
The answer is Unique Factorization Domains. Thanks Google and Wikipedia!