About vanishing of Tor of a morphism over a perfect ring?

Question

Let $R$ be an associative ring with identity. An ideal $I$ of $R$ is said to be T-nilpotent if for every left $R$-module $M$, $IM=M$ implies $M=0$ equivalently for every left $R$-module $M$, $Hom_{R}(R/I, M)=0$ implies $M=0$. The ring $R$ is said to be left (resp, right) perfect if left (resp, right) flat R-modules are projective. $R$ is called a perfect ring if it is both left and right perfect. It is known that $R$ is perfect if and only if the Jacobson radical $J$ of $R$ is T-nilpotent and the quotient ring $R/J$ is semisimple. It follows that if $R$ is perfect then a left $R$-module $M$ is flat if and only if $Tor_{1}^{R}(R/J,M)=0$. I want to know if this statement still true for morphisms of left $R$-modules , i.e, if $R$ is perfect and $\alpha:M\to N$ is a morphism of left $R$-modules, then $Tor_{1}^{R}(A,\alpha)=0$ for any right $R$-module if and only if $Tor_{1}^{R}(R/J,\alpha)=0$, where $Tor_{1}^{R}(X,\alpha): Tor_{1}^{R}(X,M)\to Tor_{1}^{R}(X,N)$ is the induced morphism of abelian groups.