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

Gyu Whan Chang; Hwankoo Kim*

doi:10.5666/KMJ.2021.61.1.23

Article

Kyungpook Mathematical Journal 2021; 61(1): 23-32

Published online March 31, 2021

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

Gyu Whan Chang, Hwankoo Kim*

Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea
e-mail : whan@inu.ac.kr

Division of Computer and Information Engineering, Hoseo University, Asan 31499, Republic of Korea
e-mail : hkkim@hoseo.edu

Received: February 22, 2020; Revised: March 18, 2020; Accepted: May 30, 2020

Abstract

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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.

Keywords: star operation, extension of a star operation to the polynomial ring, t-class group, integrally closed, *-Noetherian domain.