# TWO GENERALIZATIONS OF LCM-STABLE EXTENSIONS

• Chang, Gyu Whan (Department of Mathematics University of Incheon) ;
• Kim, Hwankoo (Department of Information Security Hoseo University) ;
• Lim, Jung Wook (Department of Mathematics Sogang University)
• 투고 : 2012.05.02
• 발행 : 2013.03.01
• 86 5

#### 초록

Let $R{\subseteq}T$ be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then $R{\subseteq}T$ is said to be LCM-stable if $(aR{\cap}bR)T=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$. Let $w_A$ be the so-called $w$-operation on an integral domain A. In this paper, we introduce the notions of $w(e)$- and $w$-LCM-stable extensions: (i) $R{\subseteq}T$ is $w(e)$-LCM-stable if $((aR{\cap}bR)T)_{w_T}=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$ and (ii) $R{\subseteq}T$ is $w$-LCM-stable if $((aR{\cap}bR)T)_{w_R}=(aT{\cap}bT)_{w_R}$ for all $0{\neq}a,b{\in}R$. We prove that LCM-stable extensions are both $w(e)$-LCM-stable and $w$-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., $P{\upsilon}MD$), then $R{\subseteq}T$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable) if and only if $R[X]{\subseteq}T[X]$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable).

#### 키워드

star-operation;LCM-stable;w-LCM-stable;w(e)-LCM-stable;PvMD;Krull domain

#### 과제정보

연구 과제 주관 기관 : University of Incheon, NRF

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#### 피인용 문헌

1. ON LCM-STABLE MODULES vol.13, pp.04, 2014, https://doi.org/10.1142/S0219498813501338