Wa`alykum As-Salaamu Warahmatullahi Wabarakaatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother, in Islam, thanks a lot for your good question which reflect your love and care about the blessed month of Ramadan. Ramadan, as a matter of fact, is a beloved month for all Muslims, for, in addition to the blessing galore that the month is characterized with, it also takes its uniqueness in being the month in which the Glorious Qur'an was revealed to the Prophet, peace and blessings be upon him.
As for the question on the permissibility of reading from the Mushaf in Prayer, the eminent Muslim scholar, Dr. `Abdul-Fattah `Ashoor, Professor of the Exegesis of the Qur'an at Al-Azhar Univ., states:
"The more one reads the Qur'an in the correct way approved by Muslim scholars, the more rewards one receives. Performing supererogatory prayer while reading from the Mushaf (the copy of the Qur'an) is permissible, for `Aishah, Mother of the Believers, used to do so with her servant who used to lead her in supererogatory prayer while reading from the Mushaf But, if reading from one's memory, without the Mushaf , leads one to concentrate and reflect on the meaning of the Qur'an and fill one with the joys of recitation, that will be better than reading form the Mushaf. This is based on the rule that reflection on the meanings of the Qur'an and memorizing it by heart is more virtuous that a mere recitation of it. May Allah help you and accept your good deeds!
But as for the obligatory prayer, one should not perform it while reading from the Mushaf, for it is sufficient for one to read even a short Surah in prayer. Moreover, there is no record indicating that such thing was done in the era of the early Muslims. So, you should read from your memory even if Surat al-Fatihah is the only chapter you committed to heart, because reciting any other Surah after the Fatihah is Sunnah and not obligatory."