Simple Rings are V-Rings?

Question

Are all simple rings V-rings?

Because the condition of be simple ring is symmetrical there in no need of specify if it is a left or right V-ring.

In a simple artinian ring, all the left modules (also right) are injective. If we lose the hypothesis of be artinian, the simple modules are injective?

If it is false, I'm looking for a simple ring (not artinian) with a simple module no injective.

The definition of left V-Ring is a ring in which all his left simple modules are injective.

Answer 1

Not all simple rings are V-rings. See

Osofsky, B. L. On twisted polynomial rings. J. Algebra 18 1971 597–607.

In the middle of page 606, an example (example b) is given of a simple domain that is not a V-ring.

Interestingly, at the end of the paper, Dr. Osofsky comments that it "seems highly unlikely" that simple rings can have both injective and noninjective simple modules.