D上的一个星型运算在多项式环D[X]上的两个扩展

Pub Date : 2021-01-01 DOI:10.5666/KMJ.2021.61.1.23
G. Chang, Hwankoo Kim
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引用次数: 0

摘要

设D是一个有商域K的积分域,X是D上的一个不定数,D上的一个* *运算,Cl * (D)是D上的* -类群。对于D的非零有限生成理想J,当J∗= D}时,D上的* w运算是一个由I∗w = {X∈K | xJ≠I定义的*运算。本文研究了D[X]上由A{∗}= P∈∗w-Max(D) ADP [X]和A[∗]= (P∈∗w-Max(D) AD[X]P [X])∩AK[X]定义的两个星形运算{∗}和[∗]。除其他事项外,我们证明当且仅当D是积分时,Cl∗(D) ~ = Cl[∗](D[X])
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Two Extensions of a Star Operation on D to the Polynomial Ring D[X]
Let D be an integral domain with quotient field K, X an indeterminate over D, ∗ a star operation on D, and Cl∗(D) be the ∗-class group of D. The ∗w-operation on D is a star operation defined by I∗w = {x ∈ K | xJ ⊆ I for a nonzero finitely generated ideal J of D with J∗ = D}. In this paper, we study two star operations {∗} and [∗] on D[X] defined by A{∗} = ⋂ P∈∗w-Max(D) ADP [X] and A [∗] = ( ⋂ P∈∗w-Max(D) AD[X]P [X]) ∩ AK[X]. Among other things, we show that Cl∗(D) ∼= Cl[∗](D[X]) if and only if D is integrally
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