Sets of Lengths. Issue 10 (1st December 2016)

• Record Type: Journal Article
• Authors: Geroldinger, Alfred
• Is Part Of: American Mathematical Monthly. Volume 123:Issue 10(2016)
• Page Start: 960
• Page End: 988
• Publication Date: 2016-12-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.4169/amer.math.monthly.123.10.960
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element a can be a product of k irreducibles and a product of l irreducibles. The set L( a ) of all possible factorization lengths of a is called the set of lengths of a, and the system consisting of all these sets L( a ) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.