Egypt was brought into the Roman empire by the defeat of Mark Antony and Cleopatra by Octavian (Augustus). Was Mark Antony ruler of the eastern half of the empire, then Egypt was effectively still independent. But Egypt's alliance him spelt the end of their independence, once Octavian had triumphed at the decisive battle of Actium in 30 BC.
And so Egypt was annexed as a Roman province. Such was its importance to the empire due to its wealth and power that it was kept under direct rule of the emperor, Roman senators in fact needing permission by the emperor even to set foot in it.