I had a cat and a rabbit that played together. It can be done. Obviously keep the rats in a cage out in the open where the cat can get use to being around them. Let the cats see you playing with them. Picking them up etc. You may want to isolate the rats from the cat when your not around just to be safe. Once you feel like the cat is comfortable around the rats start to introduce slowly in a small room if possible on the floor. Just be ready if anything happens. It's amazing how cats will just come to realize they are just part of the family. My cat would play with my rabbit and then go outside and kill a wild one. He just knew. I never really had to work at it. He just got use to the rabbit being there in the cage and the rest went pretty easy. If your cat has behavioral problems that may be a problem but if it is fairly calm it shouldn't be a problem. I had a friend that had a cockatoo that played with his cat. It will take time and just be careful and patient! Welcome to TCS! Hope this helps!
I've had cats and rats get along. Just depends on the cat. My first rat Wiffle, and cat Fluffy got along quite well, except when Wiffle was a bit wiffy and needed a bath. Fluff used to screw her nose up at the smell! Years later I took in a stray cat Wurzel, who got used to my rats. He killed wild ones though.
My mom has had a lot of rats, cats, and dogs all coexist. The rat could have the run of the house and the dogs and cats wouldn't hurt it. It certainly helps if they all grow up together, but even adults can be taught to some degree. I'm pretty sure all bets would be off if the rat got outside though.
When my cat was allowed outside she was an avid hunter. Yet when I was caring for an injured pigeon in the backyard until a rescue had open space, she knew not to kill this particular pigeon. I don't know how they know, but sometimes they just do.