How can you tell if a Nikkor lens is an FX lens? I assume if DX is not in the nomenclature that it must be an FX lens. Is that correct?
You'll want to avoid lenses marked with DX (crop cameras only).
Nikkor has made optics for various optical systems, so you should also make sure the lens is an F-mount lens. Recently there were introduced 1 series lenses, meant for Nikon 1 mirrorless camera systems; these cannot be mounted to a dSLR.
Historically, lenses for Nikon rangefinders (S-mount), Nikonos underwater cameras and Contax cameras have been produced under the Nikkor brand. Technically, those are full-frame lenses, but need an adapter to mount to a dSLR and even then will only be usable for close-range photography due to smaller flange focal length, unless the adapter has optical elements to compensate that.