What is the ruling on the prayer of one who, in the rak'ah after Al-Fatihah, recites only short surahs because he has not memorized any others?

The obligatory part of the prayer is Al-Fatiha, and it is a pillar of the prayer due to the Prophet's (peace be upon him) saying: "There is no prayer for whoever does not recite the Opening of the Book (Al-Fatiha)." So, if a woman knows only Al-Fatiha, it suffices for her, and her prayer is valid, whether it is an obligatory prayer or a supererogatory one. It is preferable for her to memorize more surahs, such as Juz' 'Amma, to recite them in prayer.