(Solved): Consider the ring \( (\mathbb{Z} \times \mathbb{Z}, *, \perp) \) with the operations defined by \[ ...

Consider the ring \( (\mathbb{Z} \times \mathbb{Z}, *, \perp) \) with the operations defined by \[ \begin{array}{l} (a, b) *(c, d)=(a+c, b+d) \\ (a, b) \perp(c, d)=(a c, b d) \end{array} \] Justify adequately which of the following subrings of \( (\mathbb{Z} \times \mathbb{Z}, *, 1) \) are ideal. a) \( I_{1}=\{(n, n): n \in \mathbb{Z}\} \) b) \( I_{2}=\{(3 n, n): n \in \mathbb{Z}\} \) c) \( I_{3}=\{(n, m): n, m \in \mathbb{Z}\} \)