Power Series Rings Over Prüfer $v$-multiplication Domains, II

Canadian mathematical bulletin

Gyu Whan Chang¹ 

[1] Department of Mathematics Education, Incheon National University, Incheon 22012, Korea

关键词: Krull domain;    P$v$MD;    multiplicatively closed set of ideals;    power series ring;   

DOI  :  10.4153/CMB-2016-046-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $D$ be an integral domain, $X^1(D)$ be the set of height-oneprime ideals of $D$,${X_{eta}}$ and ${X_{alpha}}$ betwo disjoint nonempty sets of indeterminates over $D$,$D[{X_{eta}}]$ be the polynomial ring over $D$, and$D[{X_{eta}}][![{X_{alpha}}]!]_1$ be the first typepower series ring over $D[{X_{eta}}]$.Assume that $D$ is a Prüfer $v$-multiplication domain (P$v$MD)in which each proper integral $t$-ideal has only finitely manyminimal prime ideals(e.g., $t$-SFT P$v$MDs, valuation domains, rings of Krull type).Among other things, we show that if$X^1(D) = emptyset$ or $D_P$ is a DVR for all $P in X^1(D)$,then${D[{X_{eta}}][![{X_{alpha}}]!]_1}_{D - {0}}$ is aKrull domain.We also prove that if $D$ is a $t$-SFT P$v$MD, then the completeintegral closure of $D$ is a Krull domain andht$(M[{X_{eta}}][![{X_{alpha}}]!]_1)$ = $1$ for everyheight-one maximal $t$-ideal $M$ of $D$.

【 授权许可】

Unknown