Clifford semigroups of ideals in monoids and domains

Abstract

We investigate the ideal semigroup and the ideal class semigroup built by the fractional ideals of an ideal system on a monoid or on a domain. We provide criteria for these semigroups to be Clifford semigroups or Boolean semigroups. In particular, we consider the case of valuation monoids (domains) and of Prüfer-like monoids (domains). By the way, we prove that a monoid (domain) is of Krull type if every locally principal ideal is finite. 2000 Mathematics Subject Classification: 13C18, 13F05, 20M12, 20M14, 20M25. 2. By [12, Proposition 4.8.2]. 3. It suffices to prove that J is r-locally principal. Let M ∈ r-max(D). If J ⊂ M , then J M = D M . If J ⊂ M , then Q M ∩ D = Q implies z / ∈ Q M , hence Q M ⊂ zD M and J M = zD M .

Finite character of finitely stable domains

“…Holland et al [8] have proved its validity by translating the problem into a statement on lattice ordered groups. Independently, almost at the same time, Halter-Koch [7] proved the conjecture using the language of ideal systems on cancellative commutative monoids.…”

Flat ideals and stability in integral domains

“…On the other side, we show that the t-finite character on D suffices to have that all faithfully flat ideals are invertible (Proposition 1.13). This result may be related to the Bazzoni's conjecture [5], recently proven in [23] and in [20], which states that all locally invertible (i.e., faithfully flat) ideals of a Prüfer domain are invertible if and only if the domain has the (t-)finite character on maximal ideals.…”

w-Stability and Cliffordw-Regularity of Polynomial Rings

“…The first attempt to extend the notion of Clifford regularity in the setting of star operations is due to Kabbaj and Mimouni, who considered the t-operation [26,27,28,29]. Then Halter-Koch, in the language of ideal systems, introduced Clifford * -regularity for star operations of finite type [22]. Finally, we deepened the study of stability and Clifford regularity with respect to star operations in [17,18].…”