As you mention Golan, I guess that all your torsion theories are hereditary. Let $\tau_1$ and $\tau_2$ be t.t. on Mod$(R)$ and $\phi_1:R\to R_1$, $\phi_2:R\to R_2$ the two loc. of $R$. The fact that there exists an isomorphism $\phi:R_1\to R_2$ s.t. $\phi\phi_1=\phi_1$, \phi\phi_1=\phi_2$, means that$-\otimes_RR_1$is naturally eq. to$-\otimes_RR_2$. If$\tau_1$and$\tau_2$are perfect then these torsion functors coincide with the localization functors. Thus, in such case,$M\in \mathcal T_{\tau_1}$(the torsion class of$\tau_1$) iff$M\otimes_RR_1=0$iff$M\otimes_RR_2=0$iff$M\in \mathcal T_{\tau_2}$. So$ \mathcal T_{\tau_1}=\mathcal T_{\tau_2}$, that is,$\tau_1=\tau_2$. If your torsion theories are not perfect I do not remember if$\ker(-\otimes_RR_1)=\mathcal T_1$holds true, if so you should be able to proceed as above... 1 As you mention Golan, I guess that all your torsion theories are hereditary. Let$\tau_1$and$\tau_2$be t.t. on Mod$(R)$and$\phi_1:R\to R_1$,$\phi_2:R\to R_2$the two loc. of$R$. The fact that there exists an isomorphism$\phi:R_1\to R_2$s.t.$\phi\phi_1=\phi_1$, means that$-\otimes_RR_1$is naturally eq. to$-\otimes_RR_2$. If$\tau_1$and$\tau_2$are perfect then these torsion functors coincide with the localization functors. Thus, in such case,$M\in \mathcal T_{\tau_1}$(the torsion class of$\tau_1$) iff$M\otimes_RR_1=0$iff$M\otimes_RR_2=0$iff$M\in \mathcal T_{\tau_2}$. So$ \mathcal T_{\tau_1}=\mathcal T_{\tau_2}$, that is,$\tau_1=\tau_2$. If your torsion theories are not perfect I do not remember if$\ker(-\otimes_RR_1)=\mathcal T_1\$ holds true, if so you should be able to proceed as above...