Ohio Rev. Code § 5120.45
The state shall bear the expense of the burial or cremation of an inmate who dies in a state correctional institution, if the body is not claimed for interment or cremation at the expense of friends or relatives, or is not delivered for anatomical purposes or for the study of embalming in accordance with section 1713.34 of the Revised Code. When the expense is borne by the state, interment of the person or the person's cremated remains shall be in the institution cemetery or other place provided by the state. The managing officer of the institution shall provide at the grave of the person or, if the person's cremated remains are buried, at the grave of the person's cremated remains, a metal, stone, or concrete marker on which shall be inscribed the name and age of the person and the date of death.
R.C. §5120.45