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

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发表日期：

2024年6月18日

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Hamed Ahmed

摘要

Let $A\subseteq B$ $A \subseteq B$ be an extension of integral domains, $B[\![X]\!]$ $B [[X]]$ be the power series ring over B, and $R=A + XB[\![X]\!]$ $R = A + X B [[X]]$ be a subring of $B[\![X]\!].$ $B [[X]] .$ 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 $I = u(J_1 + XJ_2[\![X]\!])$ $I = u (J_{1} + X J_{2} [[X]])$ for some $u\in qf(R),$ $u \in q f (R),$ $J_2$ $J_{2}$ an integral divisorial v-invertible ideal of B and $J_1\subseteq J_2$ $J_{1} \subseteq J_{2}$ a nonzero ideal of A.

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

摘要