A question about exact sequences of module homomorphisms.

Question

This is a part of an example in Hungerford's Algebra:

Let $R$ be a ring, $A,B$ are $R$ modules and $f:A\rightarrow B$ is an $R$-module homomorphism. Let $\operatorname{coker} f=A/\ker\,f$ and $\operatorname{coim} f=B/\operatorname{im} f$. The books says that the following sequences are exact: $$0\rightarrow \ker f\rightarrow A\rightarrow \operatorname{coim} f\rightarrow 0$$ $$0\rightarrow \operatorname{im} f\rightarrow B\rightarrow \operatorname{coker} f\rightarrow 0.$$ Then, it says that the unlabeled maps are the the obvious inclusions and projections.

I am not sure what the maps $A\rightarrow \operatorname{coim} f$ and $B\rightarrow \operatorname{coker} f$ are?

Thank you

Answer 1

The definitions of $\operatorname{coker} f$ and $\operatorname{coim} f$ are the wrong way around: $\operatorname{coker} f = B/\operatorname{im} f$ and $\operatorname{coim} f = A/\ker f$ are correct. Then $A \to \operatorname{coim} f$ and $B \to \operatorname{coker} f$ are indeed the obvious projections.