Divisorial Ideals in the Power Series Ring $$A+XB[\![X]\!]$$

Abstract

Let \(A\subseteq B\) be an extension of integral domains, \(B[\![X]\!]\) be the power series ring over B, and \(R=A + XB[\![X]\!]\) be a subring of \(B[\![X]\!].\) In this paper, we give a complete description of v-invertible v-ideals (with nonzero trace in A) of R. We show that if B is a completely integrally closed domain and I is a fractional divisorial v-invertible ideal of R with nonzero trace over A, then \(I = u(J_1 + XJ_2[\![X]\!])\) for some \(u\in qf(R),\) \(J_2\) an integral divisorial v-invertible ideal of B and \(J_1\subseteq J_2\) a nonzero ideal of A.