Wild cats that are out in the wild and not domesticated will kill rats and mice.
However most domesticated cats that are house cats and pet cats will not usually kill or harm rats or mice.
Most cats that are domesticated and live with humans do not really want much to do with mice or rats and at most they will play with the mice or rats and then let them go.
I have a cat that will run away when he sees a rat or even a mouse.
It makes me laugh but my other cat will try to catch the mice or rats but he also leaves them alone mostly.
He at most will bite them and wrestle with the rat or mouse but he eventually lets them go.
Then again I had a few cats before that would kill mice and rats.
So it's up to the cat whether or not they will kill the rat or mice.