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The proof uses the definition of exactness and the First Isomorphism Theorem. From exactness at $Z$, $\operatorname{im}(i) = \ker(p)$.

Conditions: $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$ is an exact sequence of $R$-modules.; $R$ is a commutative ring.

The proof uses the definition of exactness and the First Isomorphism Theorem. From exactness at $Z$, $\operatorname{im}(i) = \ker(p)$.

Conditions: $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$ is an exact sequence of $R$-modules.; $R$ is a commutative ring.