I need to list all Ring Homomorphisms from $R\to R$ where $R$ is a $PID$.
Let $\phi:R\to R$ be a ring homomorphism. We know submodules of free modules are free. $R$ itself is a free module. Thus, every ideal of $R$ is free module of rank $0$ or $1$. We know kernel of a ring homomorphism is an ideal thus ker $\phi \equiv R$ or $\{1\}$. First case is not possible since $\phi(1)\ne1$. Thus there is only a single Ring Homomorphism from $R$ to $R$ namely identity
Is the proof correct? if yes, how can I extend it to $\phi:R^n\to R$? Actually I have seen it written that $Hom_R(M,R)\cong R^n$ where $M$ is free module of rank $n$ but I think it should be $R^{n-1}$