An isomorphism of rings

Mar 17, 2025 4:22 PM

Let $Z [\sqrt{2}] = {a + b \sqrt{2} | a, b \in Z}$ , and let $R \subset M_{2} (Z)$ be the ring of all $2 \times 2$ matrices of the form $[\begin{matrix} a & b \\ 2 b & a \end{matrix}]$ . Prove $Z [\sqrt{2}]$ is isomorphic to $R$ .

Hint: Start by showing the map $a + b \sqrt{2} \mapsto [\begin{matrix} a & b \\ 2 b & a \end{matrix}]$ is a ring homomorphism.

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Let ${\bf Z}\left[\sqrt{2}\right]=\left\{a+b\sqrt{2}\,|\, a,b\in{ \bf Z}\right\}$, and let $R\subset \operatorname{M}_2({\bf Z})$ be the ring of all $2\times 2$ matrices of the form $\begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$. Prove ${\bf Z}\left[\sqrt{2}\right]$ is isomorphic to $R$.

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\noindent {\itshape Hint:} Start by showing the map $a+b\sqrt{2}\mapsto \begin{bmatrix} a & b \\ 2b & a\end{bmatrix}$ is a ring homomorphism.