Second, I think that your employer is ethically entitled to first refusal on any ideas you come up with, and maybe even a second as well. If they turn you down then I don't see anything unethical with going off and doing it yourself. Of course there may be legal issues: talk to a lawyer if you think there might be some.
Third, don't over-estimate your knowledge of the market and ability to sell to it. The world is littered with products which were technically better than the competition, but failed in the market place. You may be better staying put and trying to work your ideas into the existing product piecemeal.
Not running a Google search on a job candidate is actually a reasonable thing to do. The fact that the name matches with a felon, or that people have posted nasty accusations, is no evidence of anything.
Second, I think that your employer is ethically entitled to first refusal on any ideas you come up with, and maybe even a second as well. If they turn you down then I don't see anything unethical with going off and doing it yourself. Of course there may be legal issues: talk to a lawyer if you think there might be some.
Third, don't over-estimate your knowledge of the market and ability to sell to it. The world is littered with products which were technically better than the competition, but failed in the market place. You may be better staying put and trying to work your ideas into the existing product piecemeal.