Yes, it is permissible to
kill rats, which are a kind of rodents. The evidence is the following Hadith as
Al-Bukhari in his Sahih (authentic) Book of Hadith
who said that the Prophet (peace be upon him) stated:
It is not sinful of a person in the state of Ihram or otherwise to kill five kinds of animals; (Part No. 26; Page No. 206) the crow, the kite, the mouse, the scorpion, and the rabid dog.
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.