Which heir to the Rome empire was more like the old Roman Empire? Some historians argue that the Islamic empire was more Roman, while others hold that the Carolingian empire reflected Roman culture and ideals more. what various ways in which there was religious, cultural, intellectual, and political continuity between Rome and the successor states, and which state more closely resembled the fallen Roman Empire. also identify what makes each of these empires different from Rome.

The Byzantine Empire was clearly the first successor to the Roman Empire. At one point, the Roman Empire was too large and cumbersome for one ruler to govern effectively, so the empire was divided in half between a western and eastern sphere. After the fall of the Western Empire in 476 A.D., the Eastern Empire came to be called the Byzantine Empire.
Some historians have argued that the Roman Empire did not completely collapse in 476 but continued through the Byzantine Empire. This argument is compelling because the Byzantine Empire resembled its western counterpart in many ways. The Byzantine emperor ruled with absolute authority. The Byzantine Empire also had a supreme religious leader—the Patriarch—just like the Pope in the Western Empire. The Patriarch served as the official head of the Eastern Orthodox religion.
Much like the Western Empire, the Byzantine Empire constantly tried to expand its borders and territories through military conquest. This usually brought the Byzantines into conflict with Islamic peoples to its east and south. The Byzantine Empire was never as militarily successful as its western partner and faced constant encroachment (and defeat) by nomadic tribes and Islamic armies. The Byzantine Empire eventually fell to the Islamic armies in 1453, when the Byzantine capital, Constantinople, was conquered by the Ottoman Turks.
For more information see: Herrin, Judith. Byzantium: The Surviving Life of a Medieval Empire. Princeton, NJ: Princeton University Press, 2007.