# zbMATH — the first resource for mathematics

Congruence monoids. (English) Zbl 1057.13003
Let $$R$$ be a Dedekind domain with a finite ideal class-group and the finite norm property (i.e., all proper epimorphic images of $$R$$ are finite). The authors study arithmetic properties of congruence monoids in $$R$$, which in the simplest case (corresponding to the trivial sign vector) are defined as follows: If $$I$$ is a non-zero ideal of $$R$$, and $$\Gamma\neq\{0\}$$ is a multiplicatively closed subset of $$R/I$$, then the congruence monoid corresponding to $$\Gamma$$ is defined as the pre-image of $$\Gamma$$ under the canonical map $$R\to R/I$$, to which the unit element of $$R$$ has been adjoined.
A divisor theory for these monoids is constructed, and using it the authors show (theorem 3.6) that the set of lengths of factorizations of an element $$a$$ of such a monoid into irreducibles has the same structure as in the case of rings of algebraic integers, i.e., differs from a finite arithmetical progression only by a bounded number of elements, the bound being independent of $$a$$.

##### MSC:
 13A05 Divisibility and factorizations in commutative rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 11R27 Units and factorization 20M14 Commutative semigroups
Full Text: