Assuming the business that's using your exact name is also based in Ohio, you have a legitimate gripe. One of the important reasons for registering business names is to ensure that the general public isn't confused about the identity of the companies they're doing business with.

You do not need to have trademarked your name in order to be entitled to exclusive registration of that business name (through your Secretary of State). However, make sure that you're talking about the use of your EXACT name, and not something that's close. If your company is Right Way Manufacturing and the "interloper" is Right Way Bed Manufacturing, the two might peaceably co-exist.

If you're thinking, though, of giving up your company name in exchange for compensation, then definitely speak to a corporate attorney in your area about the strategy you want to use. "Cease and desist" letters carry a lot more wallop when the person signing them has "Esq" after her name.