Left–right symmetry of PCI and V-domain properties
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A V-ring has injective simple modules on the specified side; a PCI ring has injective proper cyclic modules on that side. Must right PCI imply left PCI, and must a left V-domain also be a right V-domain?
Why it matters and what counts as progress
Why it matters: These implications test left–right symmetry without the extra Ore or Noetherian hypotheses used in earlier results.
What counts as progress: A proof of either unrestricted implication, or a domain with the required one-sided injectivity properties and failure on the opposite side.
How the problem developed
AI contribution
Claude-constructed ring separates left and right PCI and V-domain properties
There exists a countable domain R that is left PCI and left V but neither right Ore, right PCI nor right V. Thus right PCI does not imply left PCI, and left V-domain does not imply right V-domain.
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Claude-constructed ring separates left and right PCI and V-domain properties