The Byzantine Empire
When the Roman Empire separated, the eastern half became the Byzantine.
Even before the initial fall of the Roman Empire, it became weak because of the choices of one emperor, Constantine. He changed the capitol from Rome, Italy, to Byzantium. He then changed the name to Constantinople, after himself. Then, the emperor Diocletian split the empire in half, so it would be easier to control and take care of.
The Byzantine Empire lasted from 330 AD until 1453 AD, for about eleven hundred years. The reasons that the empire lasted so long was because didn't get too big, it minded it's own business, and it had an organized government. It did shrink over time, but it took a long time.
When the capitol moved to the Greece area, and the Roman Empire split, the official language changed from Latin to Greek. Also, instead of practicing the Roman Catholic Church, the Byzantines created a new form of Christianity known as Eastern(Greek) Orthodox(Christian). While the Byzantine Empire was growing strong, the western half of the Roman Empire fell into the "Dark Ages".
Trade became a very important part of how others learned from the Byzantines. Russia, for example, found the Byzantine Empire very influential. They gathered things such as the Cyrillic alphabet, the Greek Orthodox religion, art work in forms of mosaics, and the architecture. The structures with dome roofs are found not only in Arabia, but also in Moscow, Russia.
During the Byzantine Empire, a set of laws was created that was very sophisticated and useful. An emperor named Justinian created his own code of laws - based on the Roman Twelve Tables. These laws became the basis for the laws in place for many European nations.
Because the Byzantines stayed strong, and even though they kind of went down their own path, they still preserved the knowledge, art, and ideas of the old Roman and Greek cultures.