Ohio Rev. Code Ann. § 1701.26

A corporation may open transfer books in any state for the purpose of transferring shares issued by it, and it may employ agents to keep the records of its shares, or to transfer or to register shares, or both, in any state, and the acts of such agents shall be binding on the corporation. The duties and liabilities of such agents shall be such as are agreed to by the corporation. If no such transfer agent is appointed by it to act in this state, the corporation shall keep an office in this state at which shares are transferable, and at which it keeps books in which are recorded the names and addresses of all shareholders and all transfers of shares.