Blackrazor wins ... probably
If you die while wearing the ring, your soul enters it
If this necrotic damage reduces you to 0 hit points, Blackrazor devours your soul.
Since being reduced to 0 hit points doesn't usually kill you, then Blackrazor would devour your soul.
PHB, p. 197:
When you drop to 0 hit points, you either die outright or fall Unconscious, as explained in the following sections.
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
If damage reduces you to 0 hit points and fails to kill you, you fall Unconscious.
It is unlikely, but technically possible that the necrotic damage from Blackrazor could instantly kill a character. I can think of a few conditions that might make this more likely:
- Low level character might have fewer than 10 maximum hit points
- A character might gain a vulnerability to necrotic damage
- A character may have had his/her hit point maximum reduced by life draining attacks from a wraith or other source
In any case, the key part of the rules regarding dropping to 0 hit points is:
When damage reduces you to 0 hit points and there is damage remaining, you die if...
So, do you drop to 0 hit points, and then die? Does Blackrazor swallow your soul?
Your DM will have to resolve this question. To the best of my knowledge, this is not explicitly outlined in the rules, nor has it been officially answered.
For guidance, your DM might consider this answer found on page 13 or the Sage Advice compendium:
If the damage from disintegrate reduces a half-orc to
0 hit points, can Relentless Endurance prevent the orc
from turning to ash?
If disintegrate reduces you to 0 hit
points, you’re killed outright, as you turn to dust. If you’re a
half-orc, Relentless Endurance can’t save you.
The Relentless Endurance racial feature, PHB p. 41:
When you are reduced to 0 hit points but not killed outright, you can drop to 1 hit point instead.
It is possible that this Sage Advice ruling indicates that even though Relentless Endurance would put you back to 1 HP, the effect of Disintegrate, triggered by being reduced to 0 hit points, is still resolved.
That would imply that, when a character is reduced to 0 hit points, you resolve all of the effects that would happen as a result. In our case, one of those is instant death, and one is from Blackrazor. After resolving both conditions, the character is dead, and his/her soul has been swallowed by Blackrazor.
The effect of the ring cannot not be resolved until after the character is dead. At that point Blackrazor has already swallowed his/her soul.
But in the absence of explicit rules or an official answer, the decision ultimately rests with the DM.