Is it permissible for the children to keep their deceased father's rented apartment, or to take financial compensation in exchange for relinquishing it, or does the apartment become the right of its owner?

If the apartment lease contract is valid and effective, then the heirs of the deceased own the usufruct of the apartment, and they are permitted to retain it, or lease it, or take compensation for relinquishing it, as long as the use is for the same purpose or less, because the death of the father does not nullify the lease. However, if the contract is invalid (such as contracts in which the lease duration was not specified), then the heirs are not permitted to retain the apartment and must return it to its owner, and they are not allowed to take compensation for relinquishing it.