Correspondence theorem for modules

Correspondence theorem (modules): Let $M$ be an $R$-module and $N\subseteq M$ a submodule. Let $\pi\colon M\to M/N$ be the quotient map. Then the assignments

• $L \mapsto L/N$ (for submodules $L\subseteq M$ with $N\subseteq L$),
• $K \mapsto \pi^{-1}(K)$ (for submodules $K\subseteq M/N$), define inverse, inclusion-preserving bijections between the set of submodules of $M$ containing $N$ and the set of submodules of the quotient module $M/N$. The inverse image operation uses the notion of preimage under $\pi$.

Under this correspondence, lattice operations are respected (e.g. intersections and sums correspond), and many structural statements about submodules “above $N$” translate into statements about submodules of $M/N$.

Proof sketch (optional): Check that $\pi^{-1}(L/N)=L$ for $N\subseteq L$, and that $\pi(\pi^{-1}(K))=K$. Monotonicity and well-definedness follow from standard properties of quotient maps.