Let $R$ be a ring, let $M$ and $N$ be left $R$-modules with zero elements $0_M$ and $0_N$, and let $K$ be an $R$-submodule of $M$. Let $M/K$ denote the quotient left $R$-module whose elements are cosets $m+K$ with $m\in M$, and whose quotient homomorphism is the map $\pi:M\to M/K$ defined by $\pi(m)=m+K$. If $f:M\to N$ is an $R$-[module homomorphism](/page/Module%20Homomorphism) whose kernel is $\ker f=\{m\in M:f(m)=0_N\}$, and if $K\subset\ker f$, then there exists a unique $R$-module homomorphism $\overline{f}:M/K\to N$ such that $f=\overline{f}\circ\pi$.