Well, this is specifically about the Qur'an, so in essence it is about the available Christian books. At the time of the supposed creation of the Qur'an Christianity had a number of theological positions and knowing which ones the Qur'an mentioned or connected to can help us, for example, date its composition. Even the mention of Jesus in sural 5 as a follower of Allah, brings in not just Judaism, of which some traditions had one of several Messiahs, I believe, but also the Manicheans. In these traditions Muhammad claimed to be the last, the final prophet bringing together all the previous ideas associated with the prophets. But, there is, compared to Judaism, few Christian references or ideas. None, from my reading, on Christian ethics although many points in the Qur'an lead to the military Christianity of the 5th, 6th century and later.
Imo, Judaism, Christianity and Islam are deeply interwoven, with respect to written, oral and legal traditions, in terms of cultural contact (and conflict). Most of the ‚erected fences‘ are ethnographic, anthropological and do also relate to the official ‚belief‘ in actual historical events.
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In his book, "The Guide for the Perplexed," Maimonides dedicated an entire section to discussing the Islamic prophet Muhammad and the teachings of Islam.