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eISSN 0454-8124
pISSN 1225-6951

### 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

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.