Which kind of ring is this? – fields, Euclidean domains, PIDs, UFDs, and integral domains, with a ring separating each class from the next.
Ideals and quotients – the isomorphism theorems, prime and maximal ideals, radicals.
Domains and factorization – prime and irreducible elements, classes of domains, Artin–Wedderburn.
Polynomial rings – rings, ideals, and the properties inherited by \(R[x]\).
Commutative algebra – Zorn’s lemma, Nakayama’s lemma, Noetherian rings, integral extensions, localization.
Number theory – norms in number fields, Euler’s totient function, and the theorems of Euler and Fermat.