who are some of the rulers during this period?
Justinian I, Constantine, Julian, Theodosius I to name a few
Byzantine Empire
Under Constantine, Christianity did not become the exclusive religion of the state but enjoyed imperial preference since he supported it with generous privileges. Constantine established the principle that emperors could not settle questions of doctrine on their own but should summon instead general ecclesiastical councils for that purpose. His convening of both the Synod of Arles and the First Council of Nicaea indicated his interest in the unity of the Church, and showcased his claim to be its head. The rise of Christianity was briefly interrupted on the accession of the emperor Julian in 361, who made a determined effort to restore polytheism throughout the empire and was thus dubbed "Julian the Apostate" by the Church. However, this was reversed when Julian was killed in battle in 363.

Theodosius I (379–395) was the last Emperor to rule both the Eastern and Western halves of the Empire.
which region was part of the eastern roman empire The extent fluctuated, but at its peak, it stretched from Byzantium up to Syria and Italy and North Africa, how would you define this type of empire? Romano-Hellenistic
wiki:16646924
Byzantine Empire [SEP] Christianization and partition of the Empire
In 330, Constantine moved the seat of the Empire to Constantinople, which he founded as a second Rome on the site of Byzantium, a city strategically located on the trade routes between Europe and Asia and between the Mediterranean and the Black Sea. Constantine introduced important changes to the Empire's military, monetary, civil and religious institutions. In regards to his economic policies he has been accused by certain scholars of "reckless fiscality", but the gold solidus he introduced became a stable currency that transformed the economy and promoted development. Under Constantine, Christianity did not become the exclusive religion of the state but enjoyed imperial preference since he supported it with generous privileges.