Splitting the t-class group

Abstract

Let D be an integral domain and S a saturated multiplicatively closed subset of D. We say that S is a splitting set if for each 0 ≠ d ϵ D, we can write d as the product d = sa, where s ϵ S and a ϵ D, with s' D ∩ aD = s' aD for all s' ϵ S. An important example of a splitting set is the multiplicatively closed set generated by a set of principal primes having the property that for each 0 ≠ d ϵ D, there is a bound on the length of a product of these primes dividing d. If S is a splitting set, then T = {0 ≠ t ϵ D | tD ∩ sD = tsD for all s ϵ S} is a saturated multiplicatively closed subset of D. We show that the map from the monoid T( D) of t-ideals of D to the cardinal product T( D S ) x c T( D t ), given by A → ( AD S , AD T ), is an order-preservin g monoid isomorphism. Moreover, the induced map Cl t ( D) → Cl t ( D S ) x Cl t ( D τ ), given by [ A] → ([ AD S ], [ AD τ ]), is an isomorphism which splits the t-class group of D. Applications and examples of this splitting are given.