Quotient of Divisible Module is Divisible
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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.
$\blacksquare$