You probably can, but you cannot equip more than 1 of the same ring.

I was wondering about that too since it's a covenant related item. Maybe if you don't stay in the Princess Guard going into NG+ since you have to reacquire covenants?

No, but you can trade the ring for two divine blessings, though I would strongly advise against it.

I didn't know this, and ended up trading her the ring, assuming I could get a new one in new game +. don't make the same mistake I did.

Try checking the chest behind Frampt in the Firelink Shrine if you... for whatever reason... end up doing this. Covenant items and things that might prove gamebreaking if you lose them (ie. Charred Orange Ring) tend to end up there

They also appear for a few other covenants

The white ring is not exclusive to covenants, it's something that indicates Miracle Resonance. This ring boosts Miracle Resonance by 1 level.

White rings could appear in anyones world anywhere, even if they arent human form or in any covenant. They are probably bugged since the Remaster though because they are way too rare.

Item description: ... and later became wife to Flame God Flann. search for flan on google

thats why i came here, thanks!

Increases radius

the suns first born ring increases the amount of damge damaging miracles such as lightning spear or wrath of the gods do.
th ring of the sun princess increases the effects of miracle synergy, which is an online mechanic that means when one player uses a miracle sometimes in another players world a white circle will show up where the first person cast that miracle and if you cast the same miracle near that white ring it boosts the effect of the miracle.

that is adorable i wish that kind of thing was more prevalent in modern games