isomorphism of rings of real and complex matrices

Note that submatrix notation (http://planetmath.org/Submatrix) will be used within this entry. Also, for any positive integer $n$, $M_{n\times n}(R)$ will be used to denote the ring of $n\times n$ matrices with entries from the ring $R$, and $R_n$ will be used to denote the following subring of $M_{2n\times 2n}(ℝ)$:

Theorem.

For any positive integer $n$, $R_n\mathrm{\cong }{M}_{n\mathrm{\times }n}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$.

Proof.

Define $\phi :R_n\to M_{n\times n}(ℂ)$ by for $A,B\in M_{n\times n}(ℝ)$.

Let $A,B,C,D\in M_{n\times n}(ℝ)$ such that . Then $A+iB=C+iD$. Therefore, $A=C$ and $B=D$. Hence, . It follows that $\phi$ is injective.

Let $Z\in M_{n\times n}(ℂ)$. Then there exist $X,Y\in M_{n\times n}(ℝ)$ such that $X+iY=Z$. Since , it follows that $\phi$ is surjective.

Let $A,B,C,D\in M_{n\times n}(ℝ)$. Then

and

It follows that $\phi$ is an isomorphism (http://planetmath.org/RingIsomorphism).∎