Since you are a permanent resident of the US, you are considered to be a resident alien, and you are subject to the same tax laws as a US citizen, even without actually acquiring citizenship.
As a resident alien, you are subject to taxes on your income worldwide. If you were to obtain ownership or interest in a business located overseas, any profits you make from that business will need to be included in the taxable income that you report each year on your US tax return. If the country where the business is located also charges you taxes on your profits, you may claim a credit for the foreign taxes you paid on that same income to help reduce or offset any tax you would owe here in the US.
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That is correct. If you have no profits then you have no income to report and no taxes to pay from that business. This would only apply in years where you actually have a profit.