Walks around unique factorization (Q2714393)

If an integral domain is not unital then the principal ideals are substituted by all multiples of the generating element. This enables us to define unique factorization in the non-unital case. The author gives the method.NEWLINENEWLINEAll integral domains of this kind can be produced in the following way: Take a unital integral domain with unique factorization. Localize it by a prime ideal. The elements of the image of this prime ideal form a non-unital integral domain with unique factorization.NEWLINENEWLINEConsider the following properties of a unital integral domain: (1) Every element has a factorization. (2) The factorization terminates in finitely many steps. (3) The factorizations of an element contain a bounded number of factors. (4) Every element has finitely many divisors, up to association. (5) The number of factors in the decompositions of an element is the same. (6) The factorization is unique. It is shown that every property implies the previous one, except that (5) yields only (3). Examples are given to show that none of these properties imply the next.