\(\displaystyle 0\rightarrow Z\stackrel{i}{\rightarrow} Z \oplus Z \oplus Z \stackrel{j}{\rightarrow} Z \rightarrow 0\)

I am thinking of it this way: If it was a short exact sequence, then i would be injective and j would be onto. Also, \(\displaystyle (Z \oplus Z \oplus Z )/ i(Z) \) would be isomorphic to \(\displaystyle Z\) .

Now \(\displaystyle i(1)= (a,b,c) \neq 0\) for some \(\displaystyle a,b,c \in Z\). Now, is it true that

\(\displaystyle i(Z)=im(i)=Z<(a,b,c)>\) ? And that \(\displaystyle (Z \oplus Z \oplus Z )/ i(Z) = Z/aZ \times Z/bz \times Z/cZ \)?If yes, how do you show this is not isomorphic to \(\displaystyle Z\) ? If not, what am I doing wrong and how do you prove my claim?