Module Isomorphism from Z4 to Z2+Z2

Question

I'm trying to find the module isomorphisms from $Z_4 $ to $ Z_2 \oplus Z_2$

I understand it has to map the zero element to zero element. But after that I don't know what to do. Also I understand this problem can be viewed as finding the group isomorhpisms.

Answer 1

There is no such isomorphism, as all elements in $\mathbf Z/2\mathbf Z\times\mathbf Z/2\mathbf Z$ have order $\le 2$, and $\mathbf Z/4\mathbf Z$ has a generator of order $4$.