Let given ring $R$ without zero divizors, where adittive group of $R$ with zero torsion. Let given subring $R_0\leq R$, and $p$ is prime number, such that $\forall r\in R, \exists i>0 : p^ir\in R_0$. Is it true that if $Nil(R_0/pR_0)=\{0\}$, then $Nil(R/pR) =\{0\}$?

Second question:

Is previous problem true in case $rk(R_0) <\infty$?