In the name of Allah, We praise Him, seek His help and ask for His forgiveness. Whoever Allah guides none can misguide, and whoever He allows to fall astray, none can guide them aright. We bear witness that there is none worthy of worship but Allah Alone, and we bear witness that Muhammad (saws) is His slave-servant and the seal of His Messengers.
Islam does not prohibit the believers from keeping pure animals or birds or fish as pets, if one indeed wishes to do so…..but Islam strictly guides that if one chooses to keep animals as pets, they should honor them and provide sufficiently for them.  If one is unable to take care of their pets, then it would be best to let them free so that they may search for their own sustenance; but if perchance one were to keep pets and does not provide sufficiently for them, that indeed would be considered a sin in the Sight of the Lord.
Sahih Al-Bukhari Hadith 3.553Â Â Â Â Â Â Â Â Â Narrated by Abdullah bin Umar
Allah’s Messenger (saws) said, “A woman was tortured and was put in Hell because of a cat which she had kept locked till it died of hunger.”
Sahih Muslim Hadith 6638AÂ Â Â Narrated by Abu Hurayrah
Allah’s Messenger (saws) said: ‘A woman was thrown into Hell-Fire because of a cat whom she had tied and did not provide it with food, nor did she set it free to eat insects of the earth until it died inch by inch.’
But if one chooses or wishes to keep lawful and pure animals as pets and subsequently honors and provides sufficiently for them, there is absolutely no harm and restriction against such an indulgence in Islam.