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.

  • $\begingroup$ Appropriated: take (something) for one's own use, typically without the owner's permission. :-) $\endgroup$ – Theo Bendit May 20 at 1:00
  • $\begingroup$ Ok,and I feel sorry about that. $\endgroup$ – user469246 May 20 at 1:01
  • $\begingroup$ You mean homomorphisms sending $1$ to $1$. I guess you know the axiom of choice implies there are plenty of automorphisms of $\Bbb{C}$. Then for $n=2$ you can embed $\Bbb{C}$ as $\Bbb{R} I +\Bbb{R}P \pmatrix{ 0 & -1\\ 1 & 0 }P^{-1}\subset M_2(\Bbb{R})$ for any $P \in GL_2(\Bbb{R})$ which gives plenty of (injective) homomorphisms $\Bbb{R} \to \Bbb{C}\to \Bbb{C} \to M_2(\Bbb{R})$. $\endgroup$ – reuns May 20 at 1:09
  • $\begingroup$ @reuns Yes, and I edited my question. $\endgroup$ – user469246 May 20 at 1:10
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    $\begingroup$ Let a ring embedding $\phi : \Bbb{C} \to M_n(\Bbb{R})$, will it be of the form $\phi = \iota \circ \sigma$ with $\sigma \in Aut(\Bbb{C})$ and $\iota(a+ib) = aI+ bJ$ for some $J^2=-I$ or is it possible that $\phi(\Bbb{C})$ will be something complicated involving the axiom of choice ? $\endgroup$ – reuns May 20 at 1:20

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