Is Keeping Cats Allowed in Islam?
Question

Is it permissible to keep a cat in the house according to Islam and its teachings?
Praise be to Allah.Is it permissible to keep cats in Islam?
It is permissible to keep cats in the house , and there is nothing wrong with that because cats are not harmful or najis (impure). 
With regard to cats not being harmful, no one disputes that, rather they are useful because they eat snakes, rats, bugs and other things that may be in the house or yard. 
Are cats impure in Islam?
With regard to cats not being najis , this is known from the hadith of Kabshah bint Ka’b ibn Malik, who said that Abu Qatadah – her husband's father – entered upon her and she poured water for him to do wudu, and a cat came to drink from it, so he tipped the vessel for it to drink. Kabshah said: “He saw me looking at him and said, “Do you find it strange, O daughter of my brother?’ I said, ‘Yes.’ He said, ‘The Messenger of Allah (peace and blessings of Allah be upon him) said, “They (cats) are not najis, rather they are of those who go around amongst you.”’” (Narrated by al-Tirmidhi, 92; al-Nasai, 68; Abu Dawood, 75; Ibn Majah, 367. classed as sahih by al-Albani. Ibn Hajar narrated in al-Talkhis that al-Bukhari classed it as sahih). 
What does Islam say about animal welfare?
It was narrated that Ibn ‘Umar (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “A woman entered Hell because of a cat which she tied up and did not feed, nor did she let it loose to feed upon the vermin of the earth.” (Narrated by al-Bukhari, 3140; Muslim, 2242). 
The vermin of the earth means insects and mice, etc. 
This hadith does not denounce the woman for keeping a cat, but it demonstrates that the woman’s sin was in not feeding it or leaving it to eat from the vermin of the earth. 
How Abu Hurayrah got his nickname?
The great Sahabi Abu Hurayrah (literally “Father of the Kitten”) was so called because he used to love cats and keep them. He became well known by this name and people forgot his real name, until the scholars disputed concerning his real name and there were nearly thirty different opinions as to what it was. Ibn ‘Abd al-Barr said in al-Isti'ab: “The most correct view is that his name was ‘Abd al-Rahman ibn Sakhr but none of them disputed that he was Abu Hurayrah.” 
And Allah knows best.