Question involving the homomorphism theorem.

Question

Let $A$ be a ring and $f:A \to A$ a surjective homomorphism. If $A/\ker f \cong A$ then is $f$ injective??

I know how to prove the other direction, but this one I don't know how to solve. Do you have any idea? Of course, if $A$ is finite then the exercise is trivial.

Answer 1

No, it is not true. Take $A=\Bbb Z\times\Bbb Z\times\Bbb Z\times\cdots$ and $f(x_1,x_2,x_3,\ldots)=(x_2,x_3,x_4,\ldots)$. Then $\ker f=\Bbb Z\times\{0\}\times\{0\}\times\cdots$, and $A/\ker f\simeq A$.