There is a theory that Egyptian Hieroglyphs were inspired by the Sumerian pictographs (proto-cuneiform).
I believe there were indeed cultural exchanges between the Sumerian and Egyptian civilizations in the early bronze age.
Egypt was the first civilization on the planet to unite a considerable territory under one rule in the proto-pharaonic era (3100-2700 BCE). Sumer wasn't really united until Uruk took over some of the city-states of the region under its control around 2600 BCE. But the materialistic culture of the Sumerians from c.3300 to 2700 BCE was indeed richer and its social structure was revolutionary.
It is debatable whether these unifications were in fact what we would refer to as a centralized authority. The first true centralized authority appeared with the Old Kingdom dynasty in Egypt c.2700 BCE, but if we look at each city-state in Sumer individually we see a very structured and hierarchical society around this time, with a militia force stationed in every city.
The concept of using military force in order to add territory to the king's realm really came about with the Akkadians some 400 years later.
Could Sumer have conquered egypt? - Historum - History Forums