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There is a known example due to Bergman (goes back to 1963) of a ring which is left primitive but not right primitive. I am wondering if there is another (preferably simpler) example of such a ring!?

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see page 159 of Ring Theory, volume 1, by L.H. Rowen.

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In the following works R. S. Irving, On the primitivity of certain Ore extensions (1979) and M. J. Crabb, J. Duncan and C. M. McGregor, On one-sided primitivity of Banach algebras (2010) alternative/simpler examples are provided. – Name Oct 15 '13 at 8:08

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