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Later, the 3rd dynasty of Ur under Ur-Nammu and Shulgi, whose power extended as far as southern Assyria, was the last great "Sumerian renaissance", but already the region was becoming more Semitic than Sumerian, with the rise in power of the Akkadian speaking Semites in Assyria and elsewhere, and the influx of waves of Semitic Martu (Amorites) who were to found several competing local powers including Isin, Larsa, Eshnunna and eventually Babylon. The last of these eventually came to dominate the south of Mesopotamia as the Babylonian Empire, just as the Old Assyrian Empire had already done so in the north from the late 21st century BC. The Sumerian language continued as a sacerdotal language taught in schools in Babylonia and Assyria, much as Latin was used in the Medieval period, for as long as cuneiform was utilized.

What was there an influx of waves of coming into Sumerian lands?
Amorites