Your question is very brief for what is a complicated situation. You do not say whether the “business” in the UK is unincorporated (i.e. a sole trader or partnership) or a limited company. You do not explain if your residential address is in France who would run the business and what you involvement would be – such as how frequently you would need to be in the UK.

The profits of a UK unincorporated business is subject to UK Income Tax and National Insurance contributions but a limited company is subject to Corporation tax on its profits plus dividends. Whether the profits of the business are subject to UK taxation would depend on whether you were “resident” or “domiciled” in the UK. These terms have precise meanings which you would need to consider in detail.

You also say that your “residential address is French” which could mean a number of things. The French system of taxation of individuals, including non-residents, centres on the household. So you would almost certainly be liable for certain French taxes wherever your business is located – Income Taxes and Wealth Taxes are most likely.

The situation you outline may well be possible but it may result in your paying more in taxes than you would if the business and you were resident in the same country.

I would advise you to talk over the position with a professional adviser who would be clear on your precise situation and offer advice accordingly.