Ohio Rev. Code § 5121.11
The state shall bear the expense of the burial or cremation of an indigent resident who dies in a state institution operated by the department of developmental disabilities under section 5123.03 of the Revised Code or 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. 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. § 5121.11