557 is typically commutative algebra and requires proficiency in ring theory including topics like classification of commutative rings (UFDs, PIDs, etc.), prime ideals and spectra, noetherian rings, and homological algebra. 557 usually follows "Introduction to Commutative Algebra" by Atiyah and Macdonald. 553 is an introductory course in abstract algebra, and is recommended for students who know (some) group theory and ring theory from undergrad. You don’t explicitly need any background for 553, but the course will be tough, especially depending on the professor. 553 usually follows “Abstract Algebra” by Dummit and Foote.

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