Show that $S$ is isomorphic to the ring $\mathbb{Z}[\sqrt3]$

abstract-algebraring-theory

Show that $S$ is isomorphic to the ring $\mathbb{Z}[\sqrt3]$ where $S
=$ { $\begin{bmatrix}a&3b\\b&a\end{bmatrix}: a,b \in \mathbb{Z}$}

The problems also says that:

Two rings are said to be isomorphic if there is a ring homomorphism between them
which is a set isomorphism.

I've already found $S$ is a commutative subring of $M_2(\mathbb{Z})$

And I set $\phi: S \to \mathbb{Z}[\sqrt3]$ as $\phi(A) \to A_{11} + \sqrt(3) A_{21}$ where $A \in S$

Then according to the definition of a set isomorphism, do I only need to prove that

  1. $\phi(A+B) = \phi(A) + \phi(B)$
  2. $\phi(AB) = \phi(A)\phi(B)$

by using my definition of $\phi$?

Best Answer

Yes; you need to prove that $\phi$ is an isomorphism of sets (i.e. a bijection), and that it is a homomorphism of rings. The most common definition of ring homomorphism also requires that $\phi(1)=1$, in addition to the two properties you already listed.

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