Tensor product of quotient and kernel

Question

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that \begin{equation} \ker(\phi_a)\otimes_R R/bR = 0. \end{equation} I have been trying for quite some time now, could someone lend me a hand with a hint?

Thank you in advance.

Answer 1

Think I solved it already. Since $R$ is a PID it is an integral domain, thus $\ker(\phi_a)=0$ and thus the tensor product is zero.