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In this video, a sequence of modules is exact if, at each middle object, the image of the incoming map equals the kernel of the outgoing map. For the sequence $X \xrightarrow{i} Z \xrightarrow{p} Y$, exactness at $Z$ means $\operatorname{im}(i) = \ker(p)$.

Conditions: The sequence consists of module homomorphisms.; Exactness is checked at each internal object (not the ends, unless specified by zeros).

In this video, a sequence of modules is exact if, at each middle object, the image of the incoming map equals the kernel of the outgoing map. For the sequence $X \xrightarrow{i} Z \xrightarrow{p} Y$, exactness at $Z$ means $\operatorname{im}(i) = \ker(p)$.

Conditions: The sequence consists of module homomorphisms.; Exactness is checked at each internal object (not the ends, unless specified by zeros).