what is Abstract Algebra

Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra ordinarily taught to children, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Elementary algebra can be taken as an informal introduction to the structures known as the real field and commutative algebra.
Abstract Algebra deals with many structures other than groups. What happens if we two operations called "addition" and "multiplication" that behave in ways simiar to the usual addition and multiplication of integers? We then have a structure that mathematicians call a ring. How about if the addition and multiplication behave like they do on fractions (so that we can divide, unlike in the integers)? This leads to a structure called a field. The investigation of these 3 types of structures (groups, rings, and fields) form the cornerstone of the field that mathematicians call Abstract Algebra. The term "abstract" refers to the perspective taken in the subject, which is very different from that of high school algebra. Rather than looking for the solutions to a particular problem, abstract algebra is interested in such questions as: When does a solution exist? If a solution does exist, is it unique? What general properties does a solution possess.