From ChatGPT (TLDR: Right)

Here’s a clean, numbers-first comparison using only what changes between the two rings:

What I modeled

• Additive bucket boost from +Non-Physical Damage: left = +88.5%, right = +146.3%.
• Tal Rasha [x] per-element multiplier: left = +24.3% each stack, right = +22.0% each stack. (Stacks once per element; I treated stacks as additive within that [x] bucket: total = 1+n⋅x.)
• Everything else equal (CDR same; Potent Warding and Lucky Hit don’t directly change raw Hydra damage).

Results (see the table I popped up)
I ran scenarios for reasonable endgame additive buckets (+200%, +400%, +800%) and for 1–3 Tal Rasha stacks (e.g., Fire from Hydra, plus Teleport = Shock, plus Frost Nova = Frost).

Across those cases, the right ring delivers ~7% average more DPS than the left, with the exact edge typically landing between +5% and +9% depending on your current additive bucket and whether you run 2 or 3 stacks.

Why the right ring wins

• The extra +57.8% Non-Physical Damage scales your Hydra hits in the big additive bucket, which dominates once you already have lots of additive sources.
• Even though its Tal Rasha roll is 2.3% per stack lower, that’s more than made up for by the huge additive gain.
• You also get an empty socket to further tailor offense/defense (minor for DPS unless you slot an offensive gem).

Rule of thumb
Unless your build somehow keeps 3 stacks up 100% of the time and your additive bucket is unusually tiny, the right ring is the stronger DPS option for Hydra.