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Encyclopedia > 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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... This article presents the essential definitions. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

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. Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. ... Please refer to Real vs. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... Please refer to Real vs. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...

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. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...

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. Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In abstract algebra, a homomorphism is a structure-preserving map. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

## History and examples GA_googleFillSlot("encyclopedia_square");

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 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. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ... Algebra (from Arabic: Ø§Ù„Ø¬Ø¨Ø±, al-ÄŸabr) is a branch of mathematics concerning the study of structure, relation and quantity. ... The term natural science as the way in which different fields of study are defined is determined as much by historical convention as by the present day meaning of the words. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ...

Examples of algebraic structures with a single binary operation are: In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...

More complicated examples include: In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

See algebraic structures for these and other examples. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... This article presents the essential definitions. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ...

## An example

Abstract algebra faciliates the study the 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 eS, such that ae = ea = a. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... This article gives an overview of the various ways to multiply matrices. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, associativity is a property that a binary operation can have. ...

In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... This is a list of important publications in mathematics, organized by field. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

## References and 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.
• Jimmie Gilbert, Linda Gilbert (2005). Elements of Modern Algebra. Thomson Brooks/Cole. ISBN 0-534-40264-X.
• R.B.J.T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann. ISBN 0-3405-4440-6.
• C. Whitehead (2002). Guide2 Abstract Algebra (2nd edition). ISBN 0-333-79447-8;.

A monograph available free online: Results from FactBites:

 S.O.S. Mathematics - Sites of Interest on the Web (1486 words) Algebra Help - An Algebra resource that "explains a wide range of algebra topics in a simple manner". Edwin H. Connell's Elements of Abstract and Linear Algebra is a foundational textbook on abstract algebra with emphasis on linear algebra. The Algebrator solves algebra problems just like your teacher: You enter a problem, and the software program solves it step-by-step, while providing clear explanations.
 Abstract algebra (655 words) Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. 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. Abstract algebra facilitates the study of properties and patterns that seemingly disparate mathematical concepts have in common.
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