showing $\psi: R\to \mathbb C$ is ring isomorphism.

Question

Below is an example from I.N. Herstein:

Let $R=\Bigg\{\left( \begin{array}{ccc} a & b \\ -b & a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of complex numbers. Define $\psi :R\to \mathbb C$ by $\psi\left( \begin{array}{ccc} a & b \\ -b & a \end{array} \right)=a+bi.$

It is asked to prove that $\psi$ is a ring isomorphism of $R$ onto $\mathbb C$ .

can anyone help me with some hint how to prove the ring isomorphism.

Answer 1

Prove that

• $\psi$ is a morphism of rings i.e.

$$\psi( M_z+M_{z'})= \psi(M_z)+\psi(M_{z'})$$ and $$\psi(M_zM_{z'})=\psi(M_z)\psi(M_{z'})$$ and $$\psi(M_1)=1$$ where $$ z=a+ib\quad;\quad M_z=\begin{pmatrix} a & b \\ -b & a \end{pmatrix} $$

• $\psi$ is injective i.e. $\ker \psi=\{M_0\}$
• $\psi$ is surjective i.e. $\operatorname{Im}\psi=\Bbb C$.