# 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 regular element of $R$.

Hence by definition $r$ is not a zero divisor of $R$.

Let $\eqclass m {} \in M / N$ be an arbitrary element represented by $m \in M$.

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

By definition of scalar multiplication on the quotient module $M / N$:

$r \eqclass m {} = \eqclass {m'} {}$

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

$\blacksquare$