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The cokernel of a module homomorphism $f:A\to B$ is the quotient module $B/\operatorname{im}(f)$. It represents the codomain modulo the image of the map.

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

The cokernel of a module homomorphism $f:A\to B$ is the quotient module $B/\operatorname{im}(f)$. It represents the codomain modulo the image of the map.

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

The cokernel of a module homomorphism $f:A\to B$ is defined as the quotient of the codomain $B$ by the image of $f$, denoted $\operatorname{coker}(f) = B/\operatorname{im}(f)$. It measures the failure of $f$ to be surjective by capturing the elements of $B$ that are not in the image of $A$.

Conditions: $f:A\to B$ is a homomorphism of $R$-modules.; $\operatorname{im}(f)$ is a submodule of $B$.

The cokernel of a module homomorphism $f:A\to B$ is defined as the quotient of the codomain $B$ by the image of $f$, denoted $\operatorname{coker}(f) = B/\operatorname{im}(f)$. It measures the failure of $f$ to be surjective by capturing the elements of $B$ that are not in the image of $A$.

Conditions: $f:A\to B$ is a homomorphism of $R$-modules.; $\operatorname{im}(f)$ is a submodule of $B$.