A collection $\Omega$ calls ring if $\forall A,B\in \Omega$ then $A\cup B$, $A\cap B$ and $A-B \in \Omega$.
Problem Give an example where $\Omega_{1}, \Omega_{2}$ are rings but $\Omega_{1}\times\Omega_{2}$ is not a ring; where $G_{1} \in \Omega_{1}\times\Omega_{2}$ is written $G_{1}=A_{1}\times B_{1}$ where $A_{1} \in \Omega_{1}$ and $B_{1} \in \Omega_{2}$, similarly for $G_{2}$
What I tried is use simple examples using a collection with numbers where fails $G_{1}-G_{2}$. The last thing I tried is $\Omega_{1}=\{\emptyset, \mathbb{N}\}$, $\Omega_{2}=\{\emptyset, \mathbb{N}, \mathbb{Z}-\{0\}, \mathbb{Z}^{-}-\{0\}\}$, where $\mathbb{Z}^{-}$ denotes the negatives integers.
$\Omega_{1},\Omega_{2}$ are rings but if I took, for example $G_{1}=\mathbb{N}\times(\mathbb{Z}-\{0\})$ and $G_{2}=\mathbb{N}\times(\mathbb{Z}^{-}-\{0\})$, but $G_{1}-G_{2}=[\mathbb{N}\times(\mathbb{Z}-\{0\})]-[\mathbb{N}\times(\mathbb{Z}^{-}-\{0\})]=\mathbb{N}\times\mathbb{N} \in \Omega_{1}\times\Omega_{2}$
So I don't know what example works, could you help me?, please.
Take $\Omega_1=\Omega_2=2^{\{0,1\}}$. Then $\{0\}\times\{0,1\}=\{(0,0),(0,1)\}\in\Omega_1\times\Omega_2$ and $\{0,1\}\times\{0\}=\{(0,0),(1,0)\}\in\Omega_1\times\Omega_2$ but their union $\{(0,0),(0,1),(1,0)\}\not\in\Omega_1\times\Omega_2$