Air Purifiers

we have a "cat room" for our 4 cats where they have litter boxes, but the room smells and often the smell lingers out into the hallway. i don’t want to leave the window open since the room is downstairs. would an air purifier tower (oe even just a fan) work to keep air cirulating and get rid of the smell?.

I have a baby and two other small children. The way my apartment building is set up, odors including smoke, escape from the hallway and into my apartment. I want my children to be able to breathe the cleaniest air possible. Thanks in advance.