Why are group theory and ring theory a part of abstract algebra?

Question

I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What does "algebra" has to do with those both subjects ? I tried to google it a little bit, but I get quite confused when I try to google the word "algebra". If I look at this page for example: http://en.wikipedia.org/wiki/Algebra_(ring_theory), then it seems like that "algebra" is a part of ring theory.

Answer 1

You are maybe confused by multiple meanings of the word algebra. The WP article you link to is about "an algebra" as the term is used in ring theory. There are also many specific types of algebra (Lie algebra, Boolean algebra) in other domains. Most of these fall within the general area of (abstract) algebra, which is a vast area of mathematics that can be somewhat characterised as dealing with reasoning based on exact equality (rather than approximative equalities and limits).

The adjective "abstract" is used to indicate that most of the time reasoning is not for one specific algebraic structure, but for a whole family of structures, with members of the family sharing the same axioms (groups, rings, fields, vector spaces are examples of such families).

Answer 2

This is probably overkill but it's (partly) because they fit into a much bigger picture known as Universal Algebra, where there's emphasis on things with operations only; for details, see A Course in Universal Algebra, by S. Burris & H. P. Sankappanavar, particularly pages 25 to 30.

Answer 3

The name "Algebra" comes from "Al-Jabr", part of the title of a book by Al-Khwarizmi. The book dealt with solutions to linear and quadratic equations, amongst other things.

Modern Algebra grew out of the need/urge to solve higher order polynomial equations - the work of Galois in that regard set up what we now call group theory. Ring theory also, I believe, grew out of the study of polynomial rings.

Although the theory that goes under the name Algebra is far more than that today, the name is preserved for historical reasons.