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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings,
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''.

# History

Before the nineteenth century, algebra meant the study of the solution of polynomial equations. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
definitions of various algebraic structures such as groups, rings, and fields. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples.

## Elementary algebra

The study of polynomial equations or
algebraic equations In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation ...
has a long history. Circa 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as
rhetorical algebra Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fu ...
and was the dominant approach up to the 16th century. Muhammad ibn Mūsā al-Khwārizmī originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to ...
's 1591
New Algebra New is an adjective referring to something recently made, discovered, or created. New or NEW may refer to: Music * New, singer of K-pop group The Boyz Albums and EPs * ''New'' (album), by Paul McCartney, 2013 * ''New'' (EP), by Regurgitator ...
, and even this had some spelled out words that were given symbols in Descartes's 1637 La Géométrie. The formal study of solving symbolic equations led
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
to accept what were then considered "nonsense" roots such as negative numbers and imaginary numbers, in the late 18th century. However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century. George Peacock's 1830 ''Treatise of Algebra'' was the first attempt to place algebra on a strictly symbolic basis. He distinguished a new symbolical algebra, distinct from the old arithmetical algebra. Whereas in arithmetical algebra $a - b$ is restricted to $a \geq b$, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as $\left(-a\right)\left(-b\right) = ab$, by letting $a=0,c=0$ in $\left(a - b\right)\left(c - d\right)=ac + bd - ad - bc$. Peacock used what he termed the principle of the permanence of equivalent forms to justify his argument, but his reasoning suffered from the
problem of induction First formulated by David Hume, the problem of induction questions our reasons for believing that the future will resemble the past, or more broadly it questions predictions about unobserved things based on previous observations. This infere ...
. For example, $\sqrt \sqrt = \sqrt$ holds for the nonnegative real numbers, but not for general complex numbers.

## Early group theory

Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic led to the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem led to the ring of integers modulo n, the multiplicative group of integers modulo n, and the more general concepts of cyclic groups and abelian groups. Klein's 1872 Erlangen program studied geometry and led to
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambi ...
s such as the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...
and the group of
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In gener ...
s. In 1874 Lie introduced the theory of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
s, aiming for "the Galois theory of differential equations". In 1976 Poincaré and Klein introduced the group of Möbius transformations, and its subgroups such as the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...
and
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of ...
, based on work on automorphic functions in analysis. The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term “group”, signifying a collection of permutations closed under composition. Arthur Cayley's 1854 paper ''On the theory of groups'' defined a group as a set with an associative composition operation and the identity 1, today called a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
. In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left
cancellation property In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...
$b\neq c \to a\cdot b\neq a\cdot c$, similar to the modern laws for a finite abelian group. Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation. Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group. Once this abstract group concept emerged, results were reformulated in this abstract setting. For example,
Sylow's theorem In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...
was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group. Otto Hölder was particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator of two elements. Burnside, Frobenius, and Molien created the representation theory of finite groups at the end of the nineteenth century. J. A. de Séguier's 1904 monograph ''Elements of the Theory of Abstract Groups'' presented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916 ''Abstract Theory of Groups''.

## Early ring theory

Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented biquaternions, Cayley introduced
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, and Grassman introduced exterior algebras.
James Cockle Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician. Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Chart ...
presented tessarines in 1848 and coquaternions in 1849.
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in h ...
introduced
split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form :q = w + xi + yj + zk where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', '' ...
s in 1873. In addition Cayley introduced group algebras over the real and complex numbers in 1854 and
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
in two papers of 1855 and 1858. Once there were sufficient examples, it remained to classify them. In an 1870 monograph, Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the
Peirce decomposition In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decomp ...
. Frobenius in 1878 and
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
in 1881 independently proved that the only finite-dimensional division algebras over $\mathbb$ were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple Lie algebras could be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over $\mathbb$ or $\mathbb$ uniquely decomposes into the direct sums of a nilpotent algebra and a semisimple algebra that is the product of some number of
simple algebra In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simpl ...
s, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called the
Wedderburn principal theorem In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
and Artin–Wedderburn theorem. For commutative rings, several areas together led to commutative ring theory. In two papers in 1828 and 1832, Gauss formulated the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathb ...
s and showed that they form a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
(UFD) and proved the
biquadratic reciprocity Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form ...
law. Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers. The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed out, $\mathbb\left(\zeta_\right)\right)$ was not a UFD. In 1846 and 1847 Kummer introduced
ideal number In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the rin ...
s and proved unique factorization into ideal primes for cyclotomic fields. Dedekind extended this in 1971 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s, a precursor of the theory of Dedekind domains. Overall, Dedekind's work created the subject of algebraic number theory. In the 1850s, Riemann introduced the fundamental concept of a Riemann surface. Riemann's methods relied on an assumption he called Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the
direct method in the calculus of variations In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relie ...
. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially M. Noether studied algebraic functions and curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It r ...
. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit in E. Noether's work. Lasker proved a special case of the Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of primary ideals. Macauley proved the uniqueness of this decomposition. Overall, this work led to the development of algebraic geometry. In 1801 Gauss introduced binary quadratic forms over the integers and defined their equivalence. He further defined the discriminant of these forms, which is an
invariant of a binary form In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables ''x'' and ''y'' that remains invariant under the special linear group acting on the variables ''x'' and ''y''. T ...
. Between the 1860s and 1890s
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms. In 1868 Gordan proved that the graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis. Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to Hilbert's basis theorem. Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by
Abraham Fraenkel Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem ...
in 1914. His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element. In addition, he had two axioms on "regular elements" inspired by work on the p-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative. Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one. In 1920, Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called ''Idealtheorie in Ringbereichen'' (''Ideal theory in rings), analyzing ascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term " Noetherian ring", and several other mathematical objects being called ''
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
''. Noted algebraist Irving Kaplansky called this work "revolutionary"; results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom. Artin, inspired by Noether’s work, came up with the descending chain condition. These definitions marked the birth of abstract ring theory.

## Early field theory

In 1801 Gauss introduced the integers mod p, where p is a prime number. Galois extended this in 1830 to finite fields with $p^n$ elements. In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893. In 1881 Leopold Kronecker defined what he called a ''domain of rationality'', which is a field of
rational fraction In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions. A ration ...
s in modern terms. The first clear definition of an abstract field was due to
Heinrich Martin Weber Heinrich Martin Weber (5 March 1842, Heidelberg, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He ...
in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry. In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today.

## Other major areas

* Solving of
systems of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...
, which led to linear algebra

## Modern algebra

The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. 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. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an ''abstract group''. Questions of structure and classification of various mathematical objects came to forefront. These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
. Hence such things as
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of
Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famou ...
and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's '' Moderne Algebra'', the two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word ''algebra'' from ''the theory of equations'' to the ''theory of algebraic structures''.

# Basic concepts

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer
nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
theorems and fewer applications. Examples of algebraic structures with a single
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary ope ...
are: * Magma *
Quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...
*
Monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
*
Semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
* Group Examples involving several operations include: * Ring * Field * Module * Vector space * Algebra over a field *
Associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplicat ...
* Lie algebra * Lattice * Boolean algebra

# Branches of abstract algebra

## Group theory

A group is a set $G$ together with a "group product", a binary operation $\cdot: G \times G \rightarrow G$. The group satisfies the following defining axioms: Identity: there exists an element $e$ such that, for each element $a$ in $G$, it holds that $e \cdot a = a \cdot e = a$. Inverse: for each element $a$ of $G$, there exists an element $b$ so that $a \cdot b = b \cdot a = e$. Associativity: for each triplet of elements $a,b,c$ in $G$, it holds that $\left(a \cdot b\right) \cdot c = a \cdot \left(b \cdot c\right)$.

## Ring theory

A ring is a set $R$ together with two binary operations, addition: $+: R \times R \rightarrow R$ and multiplication: $\cdot: R \times R \rightarrow R$. Additionally, $R$ satisfies the following defining axioms: Addition: $R$ is a commutative group under addition. Multiplication: $R$ is a monoid under multiplication. Distributive: Multiplication is distributive with respect to addition.

# Applications

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by ...
, proved in 2003, asserts that the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory,
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was ...
proved Fermat's Last Theorem. In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
s, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s interact with the force they mediate if the Lie algebra is nonabelian.

* Coding theory *
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
* List of publications in abstract algebra

* * * *