What is the Shar'i ruling on the issue of two sons building two houses at their own expense above their deceased mother's house, knowing that the land and the houses are registered in the mother's name and she did not register them to her children?

If two sons build a structure on their mother's land, it is either with the intention of it being a gift (hibah) or with the intention of ownership. If it was not with the intention of it being a gift, then they are entitled to deduct the cost of the construction from the inheritance before its distribution, provided they have evidence for that. Or, they may take an oath (perform a yameen) if the other heirs deny it and they do not possess evidence. However, the mere permission from the mother to build does not constitute transfer of ownership.