TWO GENERALIZATIONS OF LCM-STABLE EXTENSIONS

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).