From State dept:
"A U.S. national may acquire foreign nationality by marriage, or a person naturalized as a U.S. national may not lose the nationality of the country of birth. U.S. law does not mention dual nationality or require a person to choose one nationality or another. Also, a person who is automatically granted another nationality does not risk losing U.S. nationality. However, a person who acquires a foreign nationality by applying for it may lose U.S. nationality. In order to lose U.S. nationality, the law requires that the person must apply for the foreign nationality voluntarily, by free choice, and with the intention to give up U.S. nationality.
Intent can be shown by the person's statements or conduct. The U.S. Government recognizes that dual nationality exists but does not encourage it as a matter of policy because of the problems it may cause. Claims of other countries on dual national U.S. nationals may conflict with U.S. law, and dual nationality may limit U.S. Government efforts to assist nationals abroad. The country where a dual national is located generally has a stronger claim to that person's allegiance."
And from a legal website:
"Each country has its own policies regarding dual citizenship and renunciation of nationality. Although the oath of allegiance that new U. S. citizens take states that they are renouncing all other governments, often that has as much weight in their home country as a similar oath taken by an American would have in the eyes of the U. S. government. Just as those wishing to renounce U. S. citizenship must follow specific steps, so must those who are giving up another nationality."
@LisaSTl:
The official category of a green card to enter the USA is "Immigrant visa." All these student visas, H-1, fiance visa, etc are classified as "non-immigrant visas." Green cards do not expire. Just the actual plastic card is good for ten years, just like a driver's license or a credit card expires but even if you do not go to replace it and keep the expired one it does not mean that you lose your permanent residency.