Two Extensions of a Star Operation on D to the Polynomial Ring D[X]
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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 closed.
Abstract
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 closed.
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