consider a commutative unitary ring $R$ and $M$ is an $R$-module. For $ a\in R, \ \ f_a:M\rightarrow M \ \ f_a(m)=a\cdot m$ is a homomorphism. What is the kernel and cokernel of this map?
The kernel I found is $ker=\{ m\in M \ | f_a(m)=O_m\}= \{ m\in M \ | a\cdot m=O_m\}=\{O_m\} $
I am not sure about the final equality.
I know that cokernel is equal to $\frac{M}{Imf_a}$
how can I find cokernel?
how can I express cokernel of any map as a set? like for kernel $ker=\{ m\in M \ | f_a(m)=O_m\}$ how can I write cokernel? I mean $coker=\{ m+Imf_a \ | \ \ \ \ ? \ \ \ \ \}$
please help