I am claiming the following is not a short exact sequence

$\displaystyle 0\rightarrow Z\stackrel{i}{\rightarrow} Z \oplus Z \oplus Z \stackrel{j}{\rightarrow} Z \rightarrow 0$

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?