Is it permissible to keep cats as pets in the home according to Islamic teachings?

It is permissible to keep a cat in the house because it is not harmful; rather, it is beneficial by eating snakes, rats, and insects. It is also not impure, based on the hadith (definition): "It is not impure; it is one of those who frequently go around among you." The woman in the hadith you mentioned was not blamed for keeping a cat; rather, her sin was for not feeding it and leaving it to eat from the vermin of the earth. The great Companion Abu Hurairah was compassionate towards cats and kept them, which is why he was named by this appellation.