Sets of Lengths

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.

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Alfred Geroldinger

ALFRED GEROLDINGER received his MSc in mathematics from the University of Vienna, his MSc in computer science from the Vienna University of Technology, and his Ph.D. from the University of Graz. He is professor of mathematics at the University of Graz, and his research interests include commutative algebra and number theory.