$\mathbb{Z}[T]$-module and extension I consider the structure of $\mathbb{Z}[T]$-module on $\mathbb{Z}$ given by the multiplication $P \times a := P(0) \times a$. 
Now i consider a morphism of ring $\phi : \mathbb{Z}[T] \to R$, that give a structure of $\mathbb{Z}[T]$-module on $R$, i denote by $t := \phi(T) \in R$. I consider next $\mathbb{Z}\otimes_{\mathbb{Z}[T]} R$ with the structure of $R$-module. Is it true that $\mathbb{Z}\otimes_{\mathbb{Z}[T]} R \simeq R/tR$ ? 

Yes that is true, because $\mathbb Z \cong \mathbb Z[T]/(T)$ as $\mathbb Z[T]$-modules with the structure that you described. Then, we may apply the general identity $A/I\otimes_A M \cong M/IM$, which holds for $A$ a ring, $I$ an ideal and $M$ an $A$-module.
