Abstract algebra

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 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 taught in schools, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Elementary algebra can be taken to as an informal introduction to the structures known as the real field and commutative algebra.

Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Fields such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory.

Two mathematical fields that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.

History and examples

Many basic algebraic structures first emerged informally in some other field of mathematics. Axioms and primitive operations were then proposed, allowing the structure to become a part of abstract algebra. In this way abstract algebra has many fruitful connections to all other branches of mathematics.

Formal definitions of certain algebraic structures began to emerge in the 19th century. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. Hence such things as group theory and ring theory took their places in pure mathematics.

Examples of algebraic structures with a single binary operation are:

magmas,
quasigroups,
monoids, semigroups and, most important, groups.

More complicated examples include:

See algebraic structures for these and other examples.

An example

Abstract algebra faciliates the study of properties and patterns that seemingly disparate mathematical concepts have in common. For example, consider the distinct operations of function composition, f(g(x)), and of matrix multiplication, AB. These two operations have, in fact, the same structure. To see this, think about multiplying two square matrices, AB, by a one column vector, x. This defines a function equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under composition and matrices under multiplication are examples of monoids. A set S and a binary operation over S, denoted by concatenation, form a monoid if the operation associates, (ab)c = a(bc), and if there exists an e∈S, such that ae = ea = a.

References and further reading

A monograph available free online:

Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3540905782.

External links

John Beachy: Abstract Algebra On Line, Comprehensive list of definitions and theorems.
Joseph Mileti: Mathematics Museum: Abstract Algebra, A good introduction to the subject in real-life terms.

Abstract algebra

Gourt Home::Gourt :: Abstract algebra

History and examples

An example

See also

References and further reading

External links