All of these rings exist ingame; whether they show up on EL is due to MA assigning the right rarity tags:
Rings (effects gained while item is equipped)
Hermetic Ring (L)* - Left Hand - 5% Increased Run Speed
Hermetic Ring Adjusted - Left Hand - 8% Increased Run Speed
Hermetic Ring Improved - Left Hand - 10% Increased Run Speed
Hermetic Ring Modified - Left Hand - 15% Increased Run Speed
Hermetic Ring Augmented - Left Hand - 20% Increased Run Speed
Hermetic Ring Perfected - Left Hand - 25% Increased Run Speed
Athenic Ring (L)* - Left Hand - 4% Increased Evade, Dodge and Jamming Chance, 4% Trauma Mitigation
Athenic Ring Adjusted - Left Hand - 8% Increased Evade, Dodge and Jamming Chance, 4% Trauma Mitigation
Athenic Ring Improved - Left Hand - 12% Increased Evade, Dodge and Jamming Chance, 8% Trauma Mitigation
Athenic Ring Modified - Left Hand - 16% Increased Evade, Dodge and Jamming Chance, 12% Trauma Mitigation
Athenic Ring Augmented - Left Hand - 20% Increased Evade, Dodge and Jamming Chance, 16% Trauma Mitigation
Athenic Ring Perfected - Left Hand - 24% Increased Evade, Dodge and Jamming Chance, 16% Trauma Mitigation
Aeglic Ring (L)* - Right Hand - +10 Max Health - 25% Increased Regeneration
Aeglic Ring Adjusted - Right Hand - +12 Max Health - 35% Increased Regeneration
Aeglic Ring Improved - Right Hand - +14 Max Health - 35% Increased Regeneration
Aeglic Ring Modified - Right Hand - +16 Max Health - 40% Increased Regeneration
Aeglic Ring Augmented - Right Hand - +18 Max Health - 40% Increased Regeneration
Aeglic Ring Perfected - Right Hand - +20 Max Health - 40% Increased Regeneration
Ares Ring (L)* - Right Hand - 4% Faster Reload, 0.2% Increased Critical Hit Chance, 4% Increased Critical Damage
Ares Ring Adjusted - Right Hand - 6% Faster Reload, 0.4% Increased Critical Hit Chance, 6% Increased Critical Damage
Ares Ring Improved - Right Hand - 8% Faster Reload, 0.6% Increased Critical Hit Chance, 8% Increased Critical Damage
Ares Ring Modified - Right Hand - 10% Faster Reload, 0.8% Increased Critical Hit Chance, 10% Increased Critical Damage
Ares Ring Augmented - Right Hand - 12% Faster Reload, 1% Increased Critical Hit Chance, 12% Increased Critical Damage
Ares Ring Perfected - Right Hand - 14% Faster Reload, 1.2% Increased Critical Hit Chance, 14% Increased Critical Damage
Artemic Ring (L)* - Right Hand - 4% Increased Taming Chance
Artemic Ring Adjusted - Right Hand - 8% Increased Taming Chance
Artemic Ring Improved - Right Hand - 12% Increased Taming Chance
Artemic Ring Modified - Right Hand - 16% Increased Taming Chance
Artemic Ring Augmented - Right Hand - 20% Increased Taming Chance
Artemic Ring Perfected - Right Hand - 24% Increased Taming Chance