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.

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.

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?

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.