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What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would work, but this is clearly false (take a matrix ring over a field, for instance).

Benson's Representations and Cohomology I, on the other hand, claims that the Jacobson radical is in fact the intersection of all maximal two sided ideals. He defines the Jacobson radical as the intersection of the annihilators of simple R-modules, which are precisely the maximal two-sided ideals. Since this is the same as the intersection of the annihilators of the individual elements of the simple modules, then this is the same as the intersections of the maximal left (or right) ideals.

I don't see the flaw in Benson's reasoning, but I seem to recall hearing somewhere else that the Jacobson radical is not always the intersection of the maximal two-sided ideals. Who is correct here?

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Here's the flaw in Benson's reasoning: The annihilator of a simple right R-module is a <em>(right) primitive ideal</em>, which is not necessarily a maximal ideal as indicated by the answers to your question. The Jacobson radical is equal to the intersection of all maximal right ideals, and is equal to the intersection of all right primitive ideals (as well as the left-handed versions of these intersections) –  Manny Reyes Oct 27 '09 at 2:16
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Dave Benson has done a lot of interesting work, but he is most often thinking about finite groups and their (finite dimensional) group algebras. His books sometimes get too casual about the generalities, so should not be taken as a basic reference for this kind of ring theory. Jacobson himself wrote a pair of textbooks Basic Algebra which cover questions about arbitrary rings thoroughly. By now there are a number of such textbook treatments. –  Jim Humphreys Dec 27 '10 at 15:47

2 Answers 2

up vote 5 down vote accepted

Certainly every maximal ideal is the annihilator of a simple R-module, but the converse isn't true. See Exercise 4.8 in Lam's "Exercises in Classical Ring Theory" for an example.

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Thanks! That's a great counterexample. Perhaps Benson is implicitly working only with Artin rings, or perhaps it's just a mistake. –  Evan Jenkins Oct 16 '09 at 20:58

A very important example is the quotient of $U(\mathfrak{g})$ (where $\mathfrak{g}$ is a simple complex Lie algebra) by the central elements killing a finite dimensional representation. This has a unique maximal ideal (the annihilator of the finite dimensional module), but its Jacobson radical is trivial, since the annihilator of the simple highest weight module in this block whose highest weight is in the anti-dominant Weyl chamber is actually faithful. There's actually an extremely interesting poset (by inclusion) of primitive ideals sitting in between.

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