Quotient of Divisible Module is Divisible

Theorem
Let $R$ be a ring with unity.

Let $M$ be a divisible left $R$-module.

Let $N \subseteq M$ be an $R$-submodule.

Then the quotient module $M/N$ is divisible.

Proof
Let $r \in R$ be a non zero divisor.

Let $[m] \in M/N$ be an arbitrary element represented by $m \in M$.

Since $M$ is divisible, there exists some $m' \in M$, such that $m = rm'$.

By definition of the scalar multiplication on the quotient module $r[m] = [m']$.

It follows, that $M/N$ is divisible.