The ring D + XDs [x] and t-splitting sets

Abstract EN

Let D be an integral domain, 5 a multiplicatively closed subset of D, and ★ a finite character star-operation on D.

We say that S is a ★-splitting set if for each 0*de D, there exist integral ideals A and B of D with (d) = (AB)*, where A* n sD = sA* for all s e S and B* Pi S * 0.

We show that D(S) = D + XDS[X] is a PVMD (resp., GGCD domain) if and only if D is a PVMD (resp., GGCD domain) and S is a r-splitting (resp., ^-splitting) subset of D.

Let S be a f-splitting set of D and let T = {/i,A„| each A, = dtDs n D for some nonzero dt e D}.

Then D = Dsn DT.

We relate the f-operation on D to the f-operation on Dsand Dr.