This is a standard course for beginning graduate students. It covers Group Theory, basic Ring & Module theory, and bilinear forms. Group Theory: Basic concepts, isomorphism theorems, normal subgroups, Sylow theorems, direct products and free products of groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating/Symmetric groups. Basic Ring Theory: Fields, Principal Ideal Domains (PIDs), matrix rings, division algebras, field of fractions. Modules over a PID: Fundamental Theorem for abelian groups, application to linear algebra: rational and Jordan canonical form. Bilinear Forms: Alternating and symmetric forms, determinants. Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules. Finite-dimensional algebras: Simple and semisimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.
Abstract Algebra, 3rd Edition, Dummit and Foote
Standard course in Abstract Algebra for undergraduate students at the level of Rutgers Math 451