On Monoids of Ideals of Orders in Quadratic Number Fields

On the arithmetic of monoids of ideals

“…and all these monoids are factorial. In orders of Dedekind domains with finite class group, monoids of all nonzero ideals and monoids of invertible ideals have similar arithmetical properties ( [16,38,14]). In contrast to (5.1) and in contrast to orders in Dedekind domains, our conjecture (Conjecture 5.12) is that the arithmetic of the monoid I(R) is completely different from the arithmetic of I * (R) and that it is as wild as it is for Krull monoids with infinite class group and prime divisors in all classes (see Proposition 4.9.3).…”

“…In Section 2, we introduce all the required arithmetical concepts in the setting of unit-cancellative monoids. The monoid of all nonzero ideals was studied for orders in Dedekind domains with finite class group [16,38,14]. In this setting, monoids of all nonzero ideals share arithmetical finiteness properties with monoids of invertible ideals and, more generally, with Krull monoids having finite class group.…”

“…Theorem 5.8],[40, Theorem 5.8],[41, Corollary 4.6],[16, Theorem 1.1],[38, Theorem 5.13],[14, Theorem 5.10]). The following corollary characterizes when -under some additional assumptions -I * v (R) is fully elastic (compare with Proposition 4.9.1).…”