Class semigroups of integral domains

Abstract

This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherian-like settings where the Clifford and Boolean properties of S(R) coincide with (Lipman and Sally-Vasconcelos) stability conditions; a main feature is that the Clifford property forces t−locally Noetherian domains to be one-dimensional Noetherian domains. Section 5 studies the transfer of the Clifford and Boolean properties to various pullback constructions. Our results lead to new families of integral domains with Clifford or Boolean class semigroup, moving therefore beyond the contexts of integrally closed domains or Noetherian domains.

Core of ideals in integral domains

“…[2,37]). Sally and Vasconcelos [50] used stability to settle Bass' conjecture on one-dimensional Noetherian rings with finite integral closure.…”

“…Recall that a nonzero ideal I of a domain R is stable (resp., strongly stable) if it is invertible (resp., principal) in its endomorphism ring (I : I) (cf. [2,37]). One can easily show that a stable ideal I satisfies I 2 I −1 ⊆ core(I).…”

Flat ideals and stability in integral domains

“…So, the quasi-stable property generalizes the stable property (instead of strengthening it as in [25]). …”

Constituent groups of Clifford semigroups arising from t-closure

“…some authors moved from the class group to the class semigroup, first considering orders in number fields [32] and valuation domains [9], and then more general contexts [5][6][7][8]24,25]. The basic idea is to look at those domains that have Clifford class semigroup.…”

“…Let F (R), Inv(R), and P (R) denote the sets of nonzero, invertible, and nonzero principal fractional ideals of R, respectively. Under this notation, the Picard group [3,4,16], class group [10,11], t-class semigroup [26], and class semigroup [9,24,25,32] of R are defined as follows: Pic(R) := Inv(R)/P (R), Cl(R) := Inv t (R)/P (R), S t (R) := F t (R)/P (R), and S(R) := F (R)/P (R). We have the settheoretic inclusions…”