Groups and Rings

A first course in abstract algebra taught at UCR as Grupos y Anillos, following Dr. Samaria Montenegro Guzmán’s lectures. It runs in two movements, groups and then rings, each developed from the axioms up.

The group theory covers subgroups, cyclic groups and the classification of their subgroups, the permutation group \(S_{n}\) and its cycle structure, the centre and centralizers, homomorphisms, chain complexes and exact sequences, translation and conjugation, cosets and Lagrange’s theorem, normal subgroups and quotient groups, the isomorphism theorems, group actions, orbits and stabilizers, simple groups and the Sylow theorems.

The ring theory covers rings, fields and integral domains, the polynomial ring \(R[x]\) with its division algorithm and roots, ideals and the ideal generated by a set, maximal ideals and Krull’s theorem, prime ideals and the nilradical, products and quotients of rings, the correspondence theorem, ring homomorphisms and the isomorphism theorems, prime subrings and characteristic, the Frobenius homomorphism and Fermat’s little theorem, the field of fractions and its universal property, and the factorization of homomorphisms through a quotient.

These are personal notes I took as a student. They were written in Spanish and translated into English; both versions are available and you can switch between them at any point. They are shared as-is, may contain errors, and are not affiliated with the instructor.