So, as the title suggests, I should find two ordinals $\alpha$ and $\beta$ such that $\alpha + \beta = \beta + \alpha$, but $\alpha \times \beta \neq \beta \times \alpha$.
I have tried all the usual examples including $\omega$, but failed to find an adequate example.
If anyone could help me, I would greatly appreciate it.
Hint: Taking $\alpha$ and $\beta$ to be similar but not equal infinite successor ordinals should work. Try something like $\alpha = \omega + 1$ and $\beta = \omega + \omega + 1$.