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!!