What is the ruling on a Quran and Arabic language teacher dealing directly with students outside the scope of the academy where he works for a fixed salary without a contract, especially since the academy did not stipulate that he should not do so?

It is permissible for a teacher to deal directly with students to teach them outside the academy for a fee, provided that the students terminate their contract with the academy. It is not permissible for the teacher to entice students to terminate their contract with the academy in order to contract with him.