chapter  9
8 Pages

The m-Complement of a Multiplicative Set

ByDavid F. Anderson, Gyu Whan Chang

Let D be an integral domain with quotient field qf(D), nonzero elements D∗ = D\{0}, and group of units U(D). For any ∅ = S ⊆ D∗, we define the m-complement for S (in D) to be ND(S) = {x ∈ D∗|xD ∩ sD = xsD for all s ∈ S}. Then ND(S) is a saturated multiplicative subset of D. When the context is clear, we write N(S) for ND(S). As in [1], a saturated multiplicative subset S of D is called a splitting set if SN(S) = D∗. It is well-known, and easily verified, that if S is a splitting set, then N(S) is also a splitting set, S ∩ N(S) = U(D), N(N(S)) = S, and D = DS ∩DN(S). As an example, U(D) is a splitting set with N(U(D)) = D∗, and thus D∗ is a splitting set with N(D∗) = U(D). Also, if S is a splitting set, then Cl(D) = Cl(DS) ⊕ Cl(DN(S)) [1, Corollary 3.8], where Cl(D) is the t-class group of D (see [6]). In [7], we investigated when Cl(D) = Cl(DS) ⊕ Cl(DN(S)) in the case that SN(S) = D \ P for some prime ideal P of D. For other properties and applications of splitting sets, see [1], [2], [3], [5], and [9]. The m-complement of S has also been studied in [3], where they used the notation S⊥ for N(S).