Jacobson's conjecture, Morita duality and related questions

Abstract

In [J], Jacobson asked a question that gave rise to the so-called Jacobson’s Conjecture: “If R is a left noetherian ring with Jacobson radical J, then nn E mr J” = 0”. Herstein [H] and Jategaonkar [Jl ] constructed counterexamples to show that this conjecture is not true. In [J3] Jategaonkar asked if Jacobson’s conjecture holds for left noetherian rings with a left Morita duality. The main purpose of this paper is to give a negative answer to this question. In Section 1 we give the basic tools for our investigations. These are elementary facts concerning trivial extensions and generalized triangular matrix rings. Many of them are already known. In section 2 we give counterexamples to the above-mentioned question. Using some Commutative Algebra and the results in Section 1, we prove (Proposition 2.5) that a large class of generalized trangular matrix rings provides counterexamples to this question. The ring R = (8 $), where V is a maximal discrete valuation domain of rank 1 and Q is its field of quotients, is a typical ring of such a class. Of course, these rings are not local but have, at least, two distinct left maximal ideals. Next (Proposition 2.11), we provide a local counterexample. This is a suitable trivial extension ring. In Section 3 we get some more results about Jacobson’s conjecture for left noetherian rings. We show that, in some particular cases, it holds. For example, if R is a left principal ideal domain with a left Morita duality or if R is noetherian (on both sides) and RR is linearly compact in the discrete topology, then Jacobson’s conjecture holds. The last one of these facts, when R has also a left Morita duality, has already been proved, in another context and using a more elaborate approach, in [J4].