If there is no testament
The first legal heirs to inherit are the children and the spouse. If there are no children, it is the spouse, then the parents. If the parents are deceased, it is the siblings. If the deceased was single, divorced, or widowed, it is the children; if there are no children, it is the parents and siblings.
If there is a will
It is the persons named in the will who must renounce. After the renunciation of the persons named in the will, the legal heirs must also renounce if they are not the same persons as those named in the will.
An inheritance may also fall to the State when the deceased leaves no relatives of the degree entitled to inherit or when no heir is known or claims the inheritance.
It is therefore important to verify the complete devolution of the estate before concluding that all those who were required to renounce have actually done so.