(A) The bond required of an administrator by section 2109.04 of the Revised Code shall not be required in either of the following cases:

1. (1) It shall not be required of a surviving spouse to administer the deceased spouse's estate if the surviving spouse is entitled to the entire net proceeds of the estate.
2. (2) It shall not be required of an administrator to administer an estate if there is no will, if the administrator is the next of kin, and if the administrator is entitled to the entire net proceeds of the estate.

(B) The bond otherwise required by section 2109.04 of the Revised Code of an administrator shall be conditioned as follows:

1. (1) To file with the probate court within the time required by section 2115.02 of the Revised Code an inventory of all tangible and intangible personal property of the deceased that is to be administered and that comes to the administrator's possession or knowledge and an inventory of the deceased's interest in real property located in this state;
2. (2) To administer and distribute according to law all tangible and intangible personal property of the deceased, the proceeds of any action for wrongful death or of any settlement, with or without suit, of a wrongful death claim, and the proceeds of all real property in which the deceased had an interest, that is located in this state, and that is sold, when the property or proceeds have come to the possession of the administrator or to the possession of a person for the administrator;
3. (3) To render a just and true account of the administrator's administration at the times required by section 2109.301 of the Revised Code;
4. (4) To deliver the letters of administration into court if a will of the deceased is proved and allowed.