Idempotents of the Class Semigroup of a Prüfer Domain of Finite Character

The class semigroup of a commutative integral domain R is the semigroup 𝒮(R) of the isomorphism classes of the nonzero ideals of R with operation induced by multiplication. A domain is said to be of finite character if every nonzero ideal is contained only in a finite number of maximal ideals. In [ 1 ] the author proved that, if R is a Prüfer domain of finite character, then 𝒮(R) is a Clifford semigroup, i.e., it is the disjoint union of the groups associated to the idempotents.

In this paper we characterise the idempotent elements of the class semigroup of a Prüfer domain of finite character and describe a generating set for the semilattice of the idempotents.