Darklone said:
Correct me if I'm wrong. I don't have Eberron nor warforged ones, but your question tells me implicitely that Warforged cannot be healed as they are mostly constructs.
That seems to be implicit in the OP's assumptions, but it is not correct.
Warforged have the "living construct" subtype, and can be healed, though spells of the Conjuration (Healing) subschool are only 50% effective upon them.
Regenerate is a Conjuration (Healing) spell, so its hit-point healing effect would be halved, though its secondary effects would work normally.
The tricky question is whether the Ring, which radiates Conjuration magic and has
Regenerate as a prerequisite, is subject to the same limitations as spells when working upon Warforged. If it is, then the closest following of the rules would be for its 1 hitpoint of healing each 5 minutes to be halved, rounded down, to zero. Alternatively, one might more reasonably halve it by having it heal 1 hitpoint per 10 minutes.
But then again, the Ring of Regeneration is not a spell, it is a magic item, and it should thus have its listed effects upon any valid target, which in this case is any living creature. Warforged are living creatures, so the ring works normally for them.