What is the ruling on the prayer of someone who prayed Dhuhr only three rak'ahs, remembered that during Asr prayer, prostrated the two prostrations of forgetfulness at that time, and must he repeat the prayer?

If you are not prone to intrusive thoughts (waswās), and you remembered that you did not complete the Dhuhr prayer as you mentioned, then your prayer is invalid and you must repeat it. The prostration of forgetfulness (sujūd al-sahw) is not sufficient for a missed rak'ah because the pillars of prayer are not rectified by it. Every Muslim must learn the rulings of individual obligations (furūḍ al-'ayn) such as prayer and purification.