The Roman empire was, as you say, an empire. Archimedes spoke, read, wrote, and taught in Greek. He wore a beard, which was unfashionable in Rome. His neighbors, friends, and family all considered themselves Greek. There was a myriad of other cultures within Rome's borders with completely different societies.

If a Puerto Rican living in Puerto Rico invented something, we'd call it a Puerto Rican invention. If a European immigrated to the US, integrated themselves into the culture by learning English, adopting American cultural norms, and taking the oath of citizenship, we'd call that an American invention, despite the fact that both of them carry a passport that says "United States of America" on it.