Long exact sequences of Abelian groups

Question

I want examples of long exact sequences of Abelian groups, as below, with no morphism being monic. $\require{AMScd}$ \begin{CD} A@>a>>B\\ @AdAA @VVbV\\ D @<<c< C \end{CD}

I've become interested in the subject but it turns out that I can't construct these sequences. Need help.

If one of the groups is $0$ the sequence is short exact and I want to examine a sort of generalization of that. \begin{CD} A@>a>>B\\ @AdAA @VVbV\\ 0 @<<c< C \end{CD}

Answer 1

What I mean is the following. If you have an exact sequence $$\cdots X_{i-1}\to X_i\to X_{i+1}\to\cdots$$ define for each $i\in\{0,1,2,3\}$ the module $S_i$ as the direct sum of all the $X_j$ with $j\equiv i\mod 4$, and maps $S_i\to S_{i+1}$, with the indices taken modulo $4$, in the obvious way to get a diagram $\require{AMScd}$ \begin{CD} S_0@>>>S_1\\ @AAA @VVV\\ S_3 @<<< S_2 \end{CD}

Answer 2

A great comment of MarianoSuárez-Álvarez.

$\require{AMScd}$ \begin{CD} A@>a>>B\\ @AdAA @VVbV\\ 0 @<<c< C \end{CD} $$\oplus$$ \begin{CD} 0@>d'>>A'\\ @Ac'AA @VVa'V\\ C' @<<b'< B' \end{CD} $$=$$ \begin{CD} A\oplus 0@>(a,d')>>B\oplus A'\\ @A(d,c')AA @VV(b,a')V\\ 0\oplus C' @<<(c,b')< C\oplus B' \end{CD}