Is it permissible to perform a four-rak'ah obligatory prayer after completing the Friday prayer, and what is the ruling on this prayer?

Praying four rak'ahs after Jumu'ah prayer is a recommended Sunnah, not an obligatory Fard. The minimum is two rak'ahs, and the maximum is four. Some scholars are of the opinion that if one prays in the mosque, they should pray four rak'ahs, and if they pray at home, they should pray two rak'ahs. As for praying Dhuhr after Jumu'ah, it is not permissible.