If any of those two ingredients is lacking, the soul ceases to exist. I will use a very basic example. If you make a cake, you need flour, milk, sugar,etc. If you were to someway extract the flour from the cake it would no longer be a cake. Nothing more nothing less.
Death doesn't mean non-existence,...in the Hebrew the translation is closer to a state of separation.
Consider what occurred when Adam and Eve partake the fruit. They remained physically alive. They hid themselves, indicating by their conscientiousness that they remained alive in the soul, but they departed from righteousness in spirit, hence their spirit had separated from God.
Prior to their sin, they were trichotomous in anthroplogy. After the sin in the Garden, man became dichotomous, body and soul.
Jesus Christ was created as the Second Adam, perfect in body, soul and spirit. His perfect sacrifice now has provided the life blood, the spirit, as redemption for what man had separated from God.
God is a god of the Living. Our faith in Him is counted for non-meritorious righteousness which by His grace is effected for salvation by Him. After Jesus Christ, man may confess his sins through Jesus Christ, for the regeneration of that spirit and be reborn in Him.
Accordingly, The soul of the unbeliever continues to exist in the Torments. The soul of the believer upon separation from the body is face to face with the Lord.
The belief that the soul ceases to exist is really the substance of existentialism, hardly the product of Scripture.