Constant Endomorphism

Question

Is the constant map $f$:M $\longrightarrow$ M with m $\longmapsto$ a

is an endomorphism , where M is a module?

let $m ,m' \in M,$ we have $f(m)=f(m')=a $ then $f(m)+f(m')=2a $ but $f(m+m')=a$ I think it isn't an homomorphism

Answer 1

As you wrote, you must have $f(m+m')=f(m)+f(m')$, i.e, $2a=a$. This is true only when $a=0$. So, if $a\neq 0$, then as you said $f$ is not an homomorphism. On the other hand, if $a=0$, then $f$ is a homomorphism!

Another method: $f(M)$ is a module, i.e, $\{a\}$ is a module. Hence $a=0$.