Problem: Suppose $I(M)$ is injective envelope of $R-$module $M$. Prove that for each endomorphism $\varphi \colon I(M) \rightarrow I(M)$ has this property $\varphi(x)=x, \forall x \in M$ then $\varphi$ is an isomorphism.
My attempt: By the hypothesis, we just check that $\varphi$ is an injective. For $x,y \in I(M)$ we have $\varphi(x) = \varphi(y) \Rightarrow x=y$ so $\varphi$ is an injective. Is my proof correct? I have to ask because my proof was very short and simple. Please check it, thank all!