We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$.
What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique determined,too.More explicitly speaking: let $\varphi \in Aut(\mathbb{R})$ ,then$\varphi|_{\mathbb{Q}}=Id_{\mathbb{Q}}$, and for $a>b \in \mathbb{R}$,we have $\varphi(a)-\varphi(b)=\varphi(a-b)=\varphi(\sqrt{a-b})^2>0$. So if we have some $a\in \mathbb{R}$,and $\varphi(a)\neq a$,we can choose a rational number between $a$ and $\varphi(a)$, which will conclude a contradiction.
Here is my question: Is there a unique ring homomorphism from $\mathbb{R}$ to another untial ring $S$(send 1 to $1_S$)?
Especially,for $S=M_n(\mathbb{R})$ the $n\times n$ matrix over $\mathbb{R}$. (For $n=1$, it is the $Aut(\mathbb{R})$,and we have determined it.)
Any suggestion will be appreciated.