Does a divorce take place if a husband swears an oath of divorce upon his wife to observe prayer on time and to complete the recitation of the Qur’an in Ramadan, and she then fails to adhere to that?

The divorce occurred due to the husband breaking his oath. If the divorce does not complete the three (talaqs), then the husband has the right to take back his wife during her 'iddah (waiting period). The return is achieved, according to some scholars, merely by the husband having intercourse with his wife during the 'iddah. If the 'iddah expires and he has not taken her back, then a new marriage contract is necessary. Some scholars, like Ibn Taymiyyah, hold the view that divorce does not occur if the husband did not intend to bring it about with that oath, and he is obligated to offer an expiation for breaking an oath. Neglecting prayer is a great evil, but the way to maintain it is not through the use of oaths.