Non-commutative arithmetic

Abstract

The problem of determining the conditions that must be imposed upon a system having a single associative and commutative operation in order to obtain unique factorization into irreducibles has been studied by A. H. Clifford [1],1 Konig [1], and Ward [2]. The more general problem of determining similar conditions for the non-commutative case has been treated by M. Ward [1], However, the conditions given by Ward are more stringent than those satisfied by actual instances of non-commutative arithmetic, for example, quotient lattices and non-commutative polynomial theory (Ore [1, 2]). Moreover, in both of these instances the factorization is unique only up to a similarity relation, and instead of a single operation of multiplication the additional operations G. C. D. and L. C. M. are involved.2 Accordingly, we shall concern ourselves with the arithmetic of a non-commutative multiplication defined over a lattice.