Divisible modules and space of divisibility of an integral domain (Q908967)

Let R be an integral domain with quotient field \(Q\neq R\) and A a nonzero divisible R-module. Then \(R_ A=\{r/s| \quad ann_ A(r)\supset ann_ A(s)\}\) is the largest overring S of R such that A is an S-module. This paper studies the ring \(R_ A\) and introduces the space of divisibility of R, d-Space(R), which is the set of equivalence classes of simple divisible R-modules, where two simple divisible R-modules C and D are equivalent if \(R_ C=R_ D\). Then d-Space(R) may be topologized to be a zero-dimensional Hausdorff space. This construction is functorial and, moreover, d-Space(R) contains a closed subspace homeomorphic to the (abstract) Riemann surface X(R) of R with the patch topology. Also, d- Space(R) is computed for several classes of integral domains.