1. The monoid of fractional ideals of a one-dimensional domain
William Heinzer and dl
in honor of Robert Gilmer
and Joe Mott

7. Fuchs and Salce, Modules over Non-Noetherian Domains, AMS, 2001
de Souza Doering and Lequain, “Extensions of semi-local rings sharing the same group of units”, JPAA, 2002.
de Souza Doering and Lequain, “The divisibility orders of Zn “, J Alg, 1999.
Zanardo and Zannier, “The class semigroup of orders in number fields”, Math Proc Cambridge Phil Soc, 1994
Anderson (Dan), Mott and Park, “Finitely generated monoids of fractional ideals”, Comm Alg, 1993
Anderson (Dan) and Mott, “Cohen-Kaplansky domains: domains with a finite number of irreducible elements”, J Alg, 1992
Mott, “Finitely generated groups of divisibility”, Contemp Math 8, AMS, 1982.

8. Notation G(R) = Q(R)*/U(R) F(R) = monoid of fractional ideals of R
S(R) = F(R)/G(R)
(I(R) = (invertible fractionals)/G(R) )
[F&S] A semigroup is “Clifford” iff it is a union of groups, i.e., iff, for all a , there is an x for which a = a2x .

9. Example: R = k[[x3,x4,x5]]
G(R’) = G(k[[x]]) = Z
G(R) = Z x ({a + bx + cx2 : a not 0} / k*)
Prop: V = k + M , N = Mn , R = k + N :
G(R) =
Z x ({a0 + a1t + … + an-1tn-1 : a0 not 0}/ k*)

10. Example (ctnd): R = k[[x3,x4,x5]]
S(R[x2]) = { R[x2] , R[x] } , both idempotent,
so Clifford.
S(R) = { R[x2] , R[x] , R, I = (x3,x4)R } ,
I2 isomorphic to M or R[x]
but S(R) not is Clifford: IR[x] = R[x] ,
not I .

11. Example: R = k[[x3,x4]]
G(R)
= Z x ((U(k[x])/ (x6))/(U(k[x3,x4]/(x6))))
S(R) = { R , R[x5] , R[x2] , R[x], (x3,x4)R,
(…???) }
R + R(x + ax2) ? ( a = 0: (x3,x4)R )

12. Example: R = k[t(t-1),t2(t-1)] , localized
G(R’) = Z2
G(R) = Z2 x ({ at + b(1-t) : a,b not 0} / k*)
Prop: Vj = k + Mj birational DVRs, j = 1,…,n ,
M = intersection of Mj’s , R = k + M :
G(R) = Zn x (U(R’/M)/k*)
= Zn x (k* x … x k*)/k*

13. Conj: Vj birational rk 1 val rings, j = 1,…,n ,
D = intersection of Vj’s
M = intersection of m(Vj)’s
k common subfield of k(Vj)’s
R pullback of k (diagonal) in D/M :
G(R) = G(V1) x … x G(Vn) x (U(D/M)/k*)