Units also have the property that they cannot be written as the product of two non-zero non-units. There's no important reason for Units and Primes to be disjoi

Units also have the property that they cannot be written as the product of two non-zero non-units. There's no important reason for Units and Primes to be disjoint sets. Units could be considered primes. It would make some things more convenient and others less convenient. For example, a prime decomposition of an element would really be a prime decomposition and not "up to a unit"; on the other hand, a proper statement of its uniqueness would be more bulky. Ultimately mathematicians decided that it's much more convenient to not consider units to be prime. It does seem to be a matter of convention, with no deep significance to it.