
Subring of the complex
Let A the subring of the complex $\displaystyle A=\{ a + b \sqrt{2}: a,b \in \mathbb{Z}\}$
a) Prove that $\displaystyle A$ is a $\displaystyle \mathbb{Z}$module and an $\displaystyle A$module, with the product as action
b) Prove that the function $\displaystyle a + b \sqrt{2} \rightarrow a+b$ is a homomorphism of $\displaystyle \mathbb{Z}$modules from $\displaystyle A$ to $\displaystyle A$ but isnīt a homomorphism of $\displaystyle A$modules
c) Prove that, as $\displaystyle \mathbb{Z}$module, A is isomorphic to $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$
thanks!!