Four periods of Sumerian can be distinguished: Archaic Sumerian, Old or Classical Sumerian, New Sumerian, and Post-Sumerian.
Archaic Sumerian covered a period from about 3100 BC, when the first Sumerian records make their appearance, down to about 2500 BC. The earliest Sumerian writing is almost exclusively represented by texts of business and administrative character. There are also school texts in the form of simple exercises in writing signs and words. The Archaic Sumerian language is still very poorly understood, partly because of the difficulties surrounding the reading and interpretation of early Sumerian writing and partly because of the meagerness of sources.
The peoples immediately to the north of Sumer were a Semitic people, the Akkadians, who were , strongly influenced by their close contacts with the Sumerians. (Akkadian Babylon was only 10 miles away from Sumerian Kish). For instance, Akkadians used Sumerian script in writing their own language.
For a long time, the Sumerians refrained from extending their political control north to the Akkadians--and the Akkadians were long busy protecting themselves from desert nomads to the west and the mountaineers to the east--while they extended their own holdings northward along the river.
Based on the above information, I appears that Sumer was a city of Sumerians until they were controlled by Akkadians.
If there is disagreement, then please express an alternate view.
Thanks,
Gerry