On the Star Class Group of a Pullback

For the domain $R$ arising from the construction $T,M,D$, we relate the star class groups of $R$ to those of $T$ and $D$. More precisely, let $T$ be an integral domain, $M$ a nonzero maximal ideal of $T$, $D$ a proper subring of $k:=T\mathrm{/}M$, $\varphi :T\to k$ the natural projection, and let $R={\varphi }^{-1}(D)$. For each star operation $\ast$ on $R$, we define the star operation ${\ast }_{\varphi }$ on $D$, i.e., the projection'' of $\ast$ under $\phi$, and the star operation ${(\ast)}_{_{T}}$ on $T$, i.e., the extension’’ of $\ast$ to $T$. Then we show that, under a mild hypothesis on the group of units of $T$, if $\ast$ is a star operation of finite type, $0\to \Cl^{\ast_{\phi}}(D) \to \Cl^\ast® \to \Cl^{{(\ast)}{{T}}}(T)\to 0$issplitexact\mathrm{.}Inparticular,when$\ast = t_{R}$,wededucethatthesequence$ 0\to \Cl^{t_{D}}(D) {\to} \Cl^{t_{R}}® {\to}\Cl^{(t_{R}){{T}}}(T) \to 0 $issplitexact\mathrm{.}Therelationbetween${(t_{R}){{T}}}$and$t_{T}$(andbetween$\Cl^{(t_{R}){{T}}}(T)$and$\Cl^{t_{T}}(T)$) is also investigated.