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No one automatically has any rights to the decedant's property. If a person dies without a will they are said to have died "intestate." In that case the decedant's estate will be probated by the court and divided according to court order.
Contact the funeral home in the state where they are to be laid to rest and they will know exactly what to do.
they did a burial
Every state has intestate laws, that outline how a person's estate will be divided if he/she dies without a will, or intestate.
A will is probated in the last state in which the person established residency.
Jehovah's Witnesses may chose to have a regular burial service conducted by a christian minister.
You will need the services of a good lawyer for a current, correct and legal answer
Each state has a section of law known as "intestacy law" which governs how property will be disposed of when a person dies without a will. Normally, it is first distributed among relatives.
The state but they might allow relatives input, if there are living relatives and the state know of them.
In all 50 US states when a person dies intestate (no will) the state probate's estate and succession laws apply. In general, the state will place the estate into trust on the behalf of the children after debts are paid off.
If somebody dies without the benefit of life insurance and no close family to foot the bill of the burial, then the body may be taken care of by the state. This may involve donating the body to medical science (yes, it does happen, not just in science fiction or horror movies) or a cut-price burial or cremation without a headstone.
In the US when a person dies with no living next of kin their property escheats to the state.