Nonunique factorization over quotients of PIDs

Abstract

We study factorizations of elements in quotients of commutative principal ideal domains that are endowed with an alternative multiplication. This study generalizes the study of factorizations both in quotients of PIDs and in rings of single-valued matrices. We are able to completely describe the sets of factorization lengths of elements in these rings, as well as compute other finer arithmetical invariants. In addition, we provide the first example of a finite bifurcus ring.