What is the definition of the cokernel of a module homomorphism?

Conditions

• $f:A\to B$ is a homomorphism of $R$-modules.

Reasoning, step by step

1. Identify the homomorphism $f:A\to B$.
2. Find the image of $f$, $\operatorname{im}(f)$, which is a submodule of $B$.
3. Construct the quotient module $B/\operatorname{im}(f)$.
4. This quotient is defined as the cokernel of $f$, $\operatorname{coker}(f)$.
5. In the video's notation for $i:X\to Z$, $\operatorname{coker}(i) = Z/\operatorname{im}(i)$.
6. The definition is introduced to interpret the statement $\operatorname{coker}(i)\simeq Y$.

Example

The video states: 'For a generic homomorphism $f:A\to B$, the board writes $\operatorname{coker} f = B/\operatorname{im} A$. This prepares the later identification of $Z/\operatorname{im} i$ with $\operatorname{coker}(i)$.'

Common misconceptions

• Confusing cokernel with kernel.
• Thinking cokernel is a submodule of the domain.
• Assuming cokernel is the image itself.

Watch the explanation

Explore next

• How is the cokernel of a module homomorphism defined?
• Why is the cokernel defined as a quotient?
• What is the relationship between cokernel and surjectivity?