The following question is from Pinter's Book of Abstract Algebra:
List all the ring homomorphisms from $\mathbb Z_2$ to $\mathbb Z_4$; and from $\mathbb Z_3$ to $\mathbb Z_6$.
I was wondering, is this a trick question? I can't really think of any homomorphisms that go in this direction. There are some that go the other way, for instance
$$ \ f = \bigl(\begin{smallmatrix} 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 0 & 1 & 2 \end{smallmatrix}\bigr), \ $$ which is from $\mathbb Z_6$ to $\mathbb Z_3$. The problem I seem to have is that if the function maps one element to two or more different elements, then it isn't a function. If my intuition is correct, could someone outline the best way to show/prove that no such homomorphisms exist? Thanks very much.