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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