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 $k:=T/M$, $\phi: T\to k$ the natural projection, and let $R={\phi}^{-1}(D)$. For each star operation $\ast$ on $R$, we define the star operation $\ast_\phi$ 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, $0\to \Cl^{\ast_{\phi}}(D) \to \Cl^\ast(R) \to \Cl^{{(\ast)}_{_{T}}}(T)\to 0$ is split exact. In particular, when $\ast = t_{R}$, we deduce that the sequence $0\to \Cl^{t_{D}}(D) {\to} \Cl^{t_{R}}(R) {\to}\Cl^{(t_{R})_{_{T}}}(T) \to 0$ is split exact. The relation between ${(t_{R})_{_{T}}}$ and $t_{T}$ (and between $\Cl^{(t_{R})_{_{T}}}(T)$ and $\Cl^{t_{T}}(T)$) is also investigated.