Why every module $P$ is the quotient of a free module implies the existence of this exact sequence specifically?

Question

Why every module $P$ is the quotient of a free module implies the existence of this exact sequence specifically $$0 \rightarrow \ker \varphi \rightarrow F \xrightarrow{\varphi} P \rightarrow 0? $$ where $F$ is a free $R$-module?

Answer 1

Assume that $P$ is the quotient of a free module. This means that $P = F/K$ for a free module $F$, and a submodule $K$. If $i: K \rightarrow F$ is the inclusion map, and $p: F \rightarrow P$ is the canonical quotient map, then the sequence

$$0 \rightarrow K \xrightarrow{i} F \xrightarrow{p} P \rightarrow 0$$

is clearly exact.