What is the religious ruling on not including in the inheritance an apartment that a father bought for his son with a cooperative loan, and is this permissible or forbidden, and how can it be corrected if it is forbidden?

If the apartment is registered in your name and your father managed it as an owner, and you possessed it completely during your father's lifetime, then it is yours. However, if you did not possess it until after his death, then the gift was not completed, and the apartment is part of the inheritance. What you paid in installments will be returned to you from the inheritance.