
List of abstract algebra topics

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called "elementary algebra". The distinction is rarely made in more recent writings.
Basic language.
Algebraic structures are defined primarily as sets with "operations". 
Structure preserving maps called "homomorphisms" are vital in the study of algebraic objects.
There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.
Advanced concepts:
Representation theory.
Representation theory