7 - Open problems

Summary

Let A = End _BU be a simple right Goldie ring, with U_A a uniform right ideal finitely generated projective and faithful over the right Ore domain B, with T = trace _BU a least ideal of B. (Some of these requirements are redundant.)

(1) Is A similar to a domain R?

This can happen iff A contains a (finitely generated) projective uniform right ideal W.

(2) Is A ≈ R_n a full n × n matrix ring over a domain R?

Clearly yes to (2) implies yes to (1).

(3) Does A have nontrivial idempotents when A is not a domain?

If e = e² ∈ A is nontrivial, then eA is a (B,A)-progenerator, where B = eAe ≈ End eA_A. Suppose that (3) has an affirmative answer. Since B has Goldie dimension less than that of A, an induction on the Goldie dimension shows that there exists e ∈ A such that B = eAe is a domain. Then A is similar to a domain, since eA is a (B, A)-progenerator. Thus, yes to (3) implies yes to (1).

(Note added in proof: A. Zalesski has announced an example giving a negative answer.)

(4) Let R be a right Ore domain with a finitely generated projective left module W such that T = trace _RW is a least ideal of R. Can W be chosen such that W is indecomposable of rank > 1?