Ring homomorphism from ring to subring that fixes subring

Question

Let $A \le B$ be rings. Suppose that there exists a unique ring homomorphism $ \phi : B \rightarrow A$ such that $ \phi (a) = a$ for all $a \in A$. Does it follow that $A=B$?

I proved that if $B$ is an integral domain and $A \neq B$ then every element in $B \setminus A$ is transcendental over $A$:
Let $x \in B$ be integral over $A$. Let $y=x- \phi (x)$. Then $y$ is integral over $A$ and $\phi (y) =0$. Let $f \in A[T]$ be monic polynomial of minimal degree such that $f(y)=0$. Then $f$ has constant term $0$, so $f(T)=T$. Hence $x=\phi (x) \in A$.