Department of Mathematics
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In 1965 Lance Small [1] recognized that certain upper triangular matrix rings provide easy counterexamples to some problems in ring theory, in particular rings having differing left and right global dimensions, and rings which are right hereditary but not left hereditary. Herstein [2] immediately recognized that these rings also provide counterexamples to Jacobson's Conjecture.

In talking about Jacobson's Conjecture over the years, I have often emphasized to our students that it is a good idea to work through the many details of the structure of the Small examples carefully. I've now observed that these examples contradict my most optimistic views on how to resolve the remaining undertermined case of Jacobson's Conjecture, but do support the overall program I have in mind. I will therefore take several lectures to do the "student assignment", and in particular give detailed computational descriptions of the indecomposable injective modules, and the injective envelope of \( R/J(R) \), in one particular case of Small's examples.

1. Lance W. Small. An example in Noetherian rings. Proc. Nat. Acad. Sci. U.S.A., 54:1035–1036 (1965).
2. I. N. Herstein. A counterexample in Noetherian rings. Proc. Nat. Acad. Sci. U.S.A., 54:1036–1037 (1965).
3. T. G. Kucera, An overview of Jacobson's Conjecture, this Seminar series, March 03, 2015, abstract.