Short exact sequence
= Short exact sequence
A <short exact sequence> is an <exact sequence>
$$
0\longrightarrow L\xrightarrow{j}M\xrightarrow{q}N\longrightarrow0.
$$
Thus $j$ is injective, $q$ is surjective, and $\operatorname{im}j=\ker q$.
= Short exact sequence
A <short exact sequence> is an <exact sequence>
$$
0\longrightarrow L\xrightarrow{j}M\xrightarrow{q}N\longrightarrow0.
$$
Thus $j$ is injective, $q$ is surjective, and $\operatorname{im}j=\ker q$.