SPLITTING THE T-CLASS GROUP

[摘要] 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 not-equal d is-an-element-of D, we can write d as the product d = sa, where s is-an-element-of S and a is-an-element-of D, with s' D intersection aD = s'aD for all s' is-an-element-of 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 not-equal d is-an-element-of 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 not-equal t is-an-element-of D \ tD intersection sD = tsD for all s is-an-element-of 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-preserving monoid isomorphism. Moreover, the induced map Cl(t)(D) --> Cl(t)(D(s)) x Cl(t)(D(T)), given by [A] --> ([AD(s)],[AD(T)]), is an isomorphism which splits the t-class group of D. Applications and examples of this splitting are given.

[发布日期] 1991-09-10