Unique factorization in multiplicative systems

Abstract

In discussing unique factorization and ideal theory, C. C. MacDuffee [l, p. 122] cites the multiplicative system of positive integers of the form 1+7& as an example where unique factorization fails, since 792 = 22-36 = 8-99, and 8, 22, 36, 99 are all primes in the system. H. Davenport [2, p. 21] uses positive integers of the form l+ik for the same purpose, with the numerical case 693 = 9-77 = 21-33. In this paper we examine all multiplicative systems made up of arithmetic progressions, and decide the question of unique factorization. For a fixed positive integer ra, let M be a multiplicatively closed system of positive integers such that if xGA2 and y=x (mod ra), y>0, then yG.M. It will be assumed that w is the smallest positive integer which can be used to define M. For example the set M of all positive integers congruent to 1, 3, or 5 modulo 6 is also the set congruent to 1 modulo 2, and in this case ra = 2. We divide the integers 1, 2, • • • , w into two classes: the set A, 0(ra) in number, of those relatively prime to ra, and the others in set B, ra—0(ra) in number.