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?
Best Answer
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.