All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
If your husband is having relationships with female students and speaks ill of you, then he has committed a great error and is exposing himself to serious temptation and doing the opposite of what Allaah has commanded him of kindness and good companionship to his wife; Allaah, The Exalted, says (what means): {...And live with them in kindness...} [Quran 4:19] He must repent to Allaah, and we have previously underlined the conditions of sincere repentance in fatwa 86527.
We hope that he has already repented to Allaah of his sinful conduct. If it is not apparent that he has gone back to such sinful conduct, then you should not have negative assumptions of him or spy on him; both actions are serious and prohibited by Islamic law; Allaah, The Exalted, says (what means): {O you who have believed, avoid much (negative) assumption. Indeed, some assumption is sin. And do not spy...} [Quran 49:12] Please refer to fataawa 138771 and 156495.
The teachings of the Sunnah forbid the wife from asking for divorce without a valid reason. Please refer to fatwa 111953 about the valid reasons for asking for divorce. If you happened to find out that your husband has gone back to his sinful conduct and you are harmed by it, then you have the right to ask for a divorce because of the harm that has been inflicted on you. Nevertheless, it goes without saying that divorce is not always the solution. Sometimes, the best solution is to be patient and strive for reconciliation, especially given that you have been blessed with children who need the both of you as one family and would not benefit of your divorce. Repentance is sufficient for such sins to be forgiven.
So be patient with your husband, and supplicate Allaah to bless him with guidance and righteousness. Also, be keen on beautifying yourself for him so that he would not be tempted by other women. For more benefit, please refer to fatwa 85044.
Allaah knows best.