1-1 correspondence between homomorphisms and the range of the homomorphisms

Question

I am having some problems understanding a problem I have been working on. Here is the exact statement of the question: ''Let $A[T]$ be the polynomial ring over a ring $A$, and $B$ any ring. Suppose that $\phi$:A 4 $\rightarrow B$ is a given ring homomorphism; show that ring homomorphisms $\psi:A[T] \rightarrow B$ extending $\phi$ are in one-to-one correspondence with elements of $B$.'' If I am not mistaken I need to find a one-to-one correspondence between the set of all $\psi$'s and the set $B$. I observed that $\psi$ is determined only by $\psi(T^n)$ (by using the fact that $\psi$ is an extension of $\phi$, which corresponds to a sequence in $B$. But this seems wrong to me, intuitively. Am I misunderstanding something here?

Answer 1

It is not only determined by $\{\psi(T^n) | n\in \mathbb{N}\}$, but as $\psi$ is multiplicative it is already determined by $\psi(T)$. This observation yields a map $\{$homomorphisms $A[T] \to B\} \to B$. Now to construct an inverse think about the following: Does $\psi(T)=b$ yield a ring homomorphism $A[T] \to B$ for any arbitrary $b \in B$?