Let us assume $I\subset \Bbb{Z}$ and $(M^i)_{i\in I}$ is a sequence of $R$ modules with associated $R$-modules homomorphism $$f^i:M^i\rightarrow M^{i+1}$$ Then we have defined the cohomology $$H^j\left(\left(M^i\right)_{i\in I}\right):=ker(f^j)/Im(f^{j-1})$$ I know that both $ker(f^j)$ and $Im(f^{j-1})$ are submodules since $f^j$ and $f^{j-1}$ are $R$-modules homomorphisms. But now I wonder how an element in this quotient looks like. Is it similar to the quotient of a ring by an ideal? So I mean if I take $\bar x\in H^j\left(\left(M^i\right)_{i\in I}\right)$ then $x\in ker(f^j)$ and $\bar x=x+Im(f^{j-1})$.
Thanks for your help