What is the ruling on bringing a prayer rug from home to pray on it inside the the mosque when it is already carpeted?

It is permissible to take a prayer rug to the mosque as long as it is not accompanied by corrupt beliefs. This falls under the category of permissible acts. It is not a condition for the mosque to be carpeted, nor is it a condition for the worshipper to pray on a mat, as long as he prays in a clean place. This is based on the Prophet's (peace and blessings be upon him) saying: "The earth has been made for me a place of prostration and a means of purification. So, any man from my Ummah whom the time of prayer approaches, let him pray."