finitely generated contracted ideal

Question

Let $f:R\to T$ be pure ring homomorphism of Noetherian commutative ring i.e.;(for every R-module $M$ we have $f\otimes id_M:R\otimes M\to T\otimes M$ injective) and $a$ is proper ideal of $R$.
Let $a^e=aT$(extension ideal of $a$). If $t_1,\cdots t_n\in aT$ s.t $Rad(<t_1,\cdots,t_n>)=Rad(aT)$, Is it true to say that $Rad<a>=Rad<f^{-1}(t_1),\cdots,f^{-1}(t_n)>$? thanks.