2015/03/16 by Alfred Geroldinger, Geroldinger, Alfred, Wolfgang Schmid +1 · 1 citation
Mathematics · #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1503.04679
Let H be a Krull monoid with class group G such that every class contains a prime divisor. Then every nonunit a ∈ H can be written as a finite product of irreducible elements. If a=u_1 ⋅ … ⋅ u_k, with irreducibles u_1, … u_k ∈ H, then k is called the length of the factorization and the set \mathsf L (a) of all possible k is called the set of lengths of a. It is well-known that the system \mathcal L (H) = \\mathsf L (a) | a ∈ H \ depends only on the class group G. In the present paper we study the inverse question asking whether or not the system \mathcal L (H) is characteristic for the class group. Consider a further Krull monoid H' with class group G' such that every class contains a prime divisor and suppose that \mathcal L (H) = \mathcal L (H'). We show that, if one of the groups G and G' is finite and has rank at most two, then G and G' are isomorphic (apart from two well-known pairings).