________________________________
Your continued donations <http://wikimediafoundation.org/wiki/Fundraising>
keep Wikipedia running!
Abstract algebra
>From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the branch of mathematics. For other uses of
the term "algebra" see algebra (disambiguation).
Abstract algebra is the field of mathematics concerned with the study of
algebraic structures such as groups, rings, fields, modules, vector spaces,
and algebras. Many of these structures were defined formally in the
nineteenth century, and, indeed, the study of abstract algebra was motivated
by the need for more rigor in mathematics. The study of abstract algebra has
brought into full view intricacies of the logical assumptions on which the
whole of mathematics and the natural sciences are built, and today there is
scarcely a branch of mathematics which doesn't utilize the results of
algebra. What's more, in the course of study, algebraists discovered that
apparently diverse logical structures can very often be brought by analogy
to a very small core of axioms. This grants the mathematician who has
learned algebra a deep sight, and empowers him broadly.
The term abstract algebra is used to distinguish the field from "elementary
algebra" or "high school algebra", which teach the correct rules for
manipulating formulas and algebraic expressions involving real and complex
numbers, and unknowns. Abstract algebra was at times in the first half of
the twentieth century known as modern algebra.
The term abstract algebra is sometimes used in universal algebra where most
authors use simply the term "algebra".
Contents
* 1 History and examples
* 2 An example
* 3 See also
* 4 Further reading
* 5 External links
[edit]
History and examples
Historically, algebraic structures usually arose first in some other field
of mathematics, were specified axiomatically, and were then studied in their
own right in abstract algebra. Because of this, abstract algebra has
numerous fruitful connections to all other branches of mathematics.
Examples of algebraic structures with a single binary operation are:
* magmas,
* quasigroups,
* monoids, semigroups and, most important, groups.
More complicated examples include:
* rings and fields
* modules and vector spaces
* algebras over fields
* associative algebras and Lie algebras
* lattices and Boolean algebras
In universal algebra, all those definitions and facts are collected that
apply to all algebraic structures alike. All the above classes of objects,
together with the proper notion of homomorphism, form categories, and
category theory frequently provides the formalism for translating between
and comparing different algebraic structures.
[edit]
An example
The systematic study of algebra has allowed mathematicians to bring under a
common logical description apparently disparate conceptions. For example,
consider two rather distinct operations: the composition of functions,
f(g(x)), and the multiplication of matrices, AB. These two operations are,
in fact, the same. To see this, think about multiplying two square matrices
(AB) by a one-column vector, x. This, in fact, defines a function that is
equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under
composition and matrices under multiplication form sets called monoids; a
monoid under an operation is associative for all its elements ( (ab)c =
a(bc) ) and contains an element e such that, for any a, ae = ea = a.
[edit]
See also
* Important publications in abstract algebra
[edit]
Further reading
* Sethuraman, B. A. (1996). Rings, Fields, Vector Spaces, and Group
Theory: An Introduction to Abstract Algebra via Geometric Constructibility,
Springer. ISBN 0-387-94848-1.
[edit]
External links
Wikibooks
<http://upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Wikibooks-logo-en.
svg/50px-Wikibooks-logo-en.svg.png>
Wikibooks has more about this subject:
Abstract algebra
<http://en.wikibooks.org/wiki/Special:Search/Abstract_algebra>
* John Beachy: Abstract Algebra On Line
<http://www.math.niu.edu/~beachy/aaol/contents.html> , Comprehensive list of
definitions and theorems.
* Joseph Mileti: Mathematics Museum: Abstract Algebra
<http://www.math.uchicago.edu/~mileti/museum/algebra.html> , A good
introduction to the subject in real-life terms.
_______________________________________________
Marxism-Thaxis mailing list
Marxism-Thaxis@lists.econ.utah.edu
To change your options or unsubscribe go to:
http://lists.econ.utah.edu/mailman/listinfo/marxism-thaxis
