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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
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