"I was married nearly eight years ago to an American citizen.  I received my green card and citizenship within five years of the marriage. Not long after we divorced.  I am still living in the U.S., and have a strong relationship with a woman in Ireland.  We are talking marriage, but I’m concerned because I’m not sure if I will be able to sponsor her for a green card. I have heard and read different things about people like myself who got legal status through an American spouse, then divorced and eventually remarrying a foreigner. Can I sponsor her for a green card?”

Yes you can.  The law says that people like yourself who receive legal status through marriage to a U.S. citizen or permanent resident can petition for a new spouse if more than five years have passed since you received a green card.  This holds true in your case.  

If five years haven’t passed, a legal spouse can also petition for a new spouse if it can be proved that the first marriage which conferred legal status was entered into for genuine reasons as opposed to immigration ones.

Those with recent divorces who quickly remarry and petition for legal status for a new spouse can expect a thorough line of questioning from immigration officials who could suspect marriage fraud.  In your case, as eight years  have elapsed, your new wife should have no problems going through the process as long as she’s otherwise qualified.