# Rings isomorphic to a proper subring

Is there a theory for rings which are isomorphic to a proper subring? Which of the following rings have this property?

$$\mathbb{R} , M_2(\mathbb{R}) , \mathbb{C} \; and \; M_2(\mathbb{Z})$$

I know that $$\mathbb{C}$$ is isomorphic to a proper subring of $$\mathbb{C}$$, assuming the axiom of choice. I have no idea about $$\mathbb{R}$$ and $$M_2(\mathbb{R})$$. I don't know about any study about such rings either. But here is a proof that $$\mathbb{C}$$ has a proper subring isomorphic to itself.
Let $$\mathcal{B}$$ be a transcendental basis of $$\mathbb{C}$$ over $$\mathbb{Q}$$. Then, note that $$\mathcal{B}$$ and $$\mathcal{B}\cup\{x\}$$ are equinumerous, where $$x$$ is a transcendental variable. Therefore, a bijection $$f:\mathcal{B}\cup\{x\}\to \mathcal{B}$$ lifts to a field isomorphism $$\varphi:\mathbb{Q}\big(\mathcal{B}\cup\{x\}\big)\to \mathbb{Q}(\mathcal{B})$$, which then can be extended to an isomorphism $$\Phi:\overline{\mathbb{Q}\big(\mathcal{B}\cup\{x\}\big)}\to \overline{\mathbb{Q}(\mathcal{B})}$$. That is, $$\Phi:\overline{\mathbb{C}(x)}\to \mathbb{C}$$ is an isomorphism of fields. Now, consider the canonical injection $$\iota:\mathbb{C}\to\overline{\mathbb{C}(x)}$$. Then, we see that the image $$S$$ of $$\mathbb{C}$$ under the composition $$\Phi\circ \iota$$ is a proper subring of $$\mathbb{C}$$ isomorphic to $$\mathbb{C}$$.
• And why do you think this wont work for $\mathbb{R}$? – Paul K Oct 17 '18 at 7:48
• Because I don't know what kind of extensions I should take to get from $\mathbb{Q}(\mathcal{T})$ to $\mathbb{R}$, if $\mathcal{T}$ is a transcendental basis of $\mathbb{R}$ over $\mathbb{Q}$. For $\mathbb{C}$ it is easy, just take the algebraic closure. Plus, since an algebraic closure is a splitting field, this allows us to extend $\varphi$ to $\Phi$. – user593746 Oct 17 '18 at 7:50
• Shouldn't it work if you exchange $\mathbb{Q}$ by its algebraic closure and for $\mathbb{R}$ you intersect it with $\mathbb{R}$? Maybe I'm missing something. – Paul K Oct 17 '18 at 7:54