Chapter 1 - Rings and ideals: we develop the basic theory of (not necessarily commutative) rings. Topics: posets and Zorn's Lemma, rings and ideals, quotients, prime and maximal ideals, Noetherian rings, the Chinese remainder theorem.
Chapter 2: UFD: we develop the classical theory of unique factorisation domains, principal ideal domains and euclidean domains.
Chapter 3 - Modules: basic notions, projective and injective modules, tensor product and flat modules, finitely generated modules, exact rows and diagram chasing, modules over principal ideal domains.
Chapter 4 - Commutative rings: we develop the theory in close connection with the elements of algebraic geometry. Topics include: the spectrum, Hilbert's Nulstellensatz, local rings.