Rings isomorphic to a proper subring

Question

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

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

Answer 1

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 $\mathbb{C}$.

Answer 2

1. You should look at A ring isomorphic to a proper subring of itself
2. If you have enough background in the topics of algebra, then you should read https://www.researchgate.net/publication/321183071_Rings_isomorphic_to_their_nontrivial_subrings