Let \(R\) be a ring and \(M\) an \(R\)-module.
For an \(R\)-module homomorphism \(\phi\colon M \to M'\), there is an isomorphism \(M/\ker\phi \cong \im\phi\).
For submodules \(A, B \subseteq M\), \[ \frac{A+B}{B} \cong \frac{A}{A \intersect B}. \]
For submodules \(A \subseteq B \subseteq M\), \[ \frac{M/A}{B/A} \cong \frac{M}{B}. \]
For a submodule \(N \subseteq M\), there is an inclusion-preserving bijection \[ \begin{aligned} \correspond{\text{Submodules of } M \\ \text{containing } N} &\mapstofrom \correspond{\text{Submodules of } M/N} \\ A &\mapstofrom A/N. \end{aligned} \]
The bijection in (d) preserves sums and intersections: for submodules \(A, A' \subseteq M\) containing \(N\), \((A+A')/N = A/N + A'/N\) and \((A \intersect A')/N = A/N \intersect A'/N\).