I have a question that I have been completely stuck on for around a week now.
Let $R$ be a ring. For $a\in A$ define a ring homomorphism $\phi_a : R[T]\rightarrow R : P(T)\mapsto P(a)$ as the evaluation at $a$. By restriction of scalars, every $\phi_a$ gives the target $R$ the structure of an $R[T]$-module, which we will denote $R_a$. Show that for $a,b\in R$, there is an $R[T]$-module isomorphism between $R_a$ and $R_b$ if and only if $a=b$.
The reverse inclusion is fine. If $a=b$, then $R_a=R_b$, so the identity map would give the required isomorphism (I'm not sure whether it would be beneficial to be more formal here?).
However, for the forward inclusion, I am completely stumped. I have deduced that $\mathrm{Im}(\phi_a)=R$ and $\ker(\phi_a) = \{P(T)\in R[T]\mid P(a)=0\}$ - that is all polynomials in $R[T]$ that have a root at $a\in R$. And was looking to see if the Isomorphism Theorem would help me, but it would seem not.
I feel that I'm missing something completely obvious here, but I've been staring for a week and nothing has presented itself. Knowledge at this stage is limited to elementary ring theory, and knowledge of basic module theory (definition, submodules, morphisms of modules, isomorphism theorems and definitions of product modules and free modules.) If anybody could offer a hint (or something to go and read) to put me on the right track, I would really appreciate it.
Thanks in advance,
Andy.