3. In mathematics, more specifically algebra, abstract algebra or modern algebra is the
study of algebraic structures. Algebraic structures include groups, rings, fields,
modules, vector spaces, lattices, and algebras over a field. The term abstract algebra
was coined in the early 20th century to distinguish it from older parts of algebra, and
more specifically from elementary algebra, the use of variables to represent numbers in
computation and reasoning. The abstract perspective on algebra has become so
fundamental to advanced mathematics that it is simply called "algebra", while the term
"abstract algebra" is seldom used except in pedagogy. Algebraic structures, with their
associated homomorphisms, form mathematical categories. Category theory gives a
unified framework to study 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.
4. Before the nineteenth century, algebra was defined as the study of polynomials.
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 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.
5. 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.[citation
needed] 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. Hence
such things as group theory 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 on the work of Ernst
Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative
rings, and of Georg Frobenius 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 reoriented the idea
MODERN ALGEBRA
6. Algebraic structures in Abstract Algebra
In abstract algebra, a group is a set of elements defined with an operation that integrates
any two of its elements to form a third element satisfying four axioms. These axioms to be
satisfied by a group together with the operation are; closure, associativity, identity and
invertibility and are called group axioms.
GROUP
7. RINGS
In mathematics, rings are algebraic structures that generalize fields: multiplication need
not be commutative and multiplicative inverses need not exist. Informally, a ring is a set
equipped with two binary operations satisfying properties analogous to those of addition
and multiplication of integers. Ring elements may be numbers such as integers or
complex numbers, but they may also be non-numerical objects such as polynomials,
square matrices, functions, and power series. Formally, a ring is a set endowed with two
binary operations called addition and multiplication such that the ring is an abelian group
with respect to the addition operator, and the multiplication operator is associative, is
distributive over the addition operation, and has a multiplicative identity element. (Some
authors define rings without requiring a multiplicative identity and instead call the
structure defined above a ring with identity.
8. FIELDS
In mathematics, a field is a set on which addition, subtraction, multiplication, and division
are defined and behave as the corresponding operations on rational and real numbers. A
field is thus a fundamental algebraic structure which is widely used in algebra, number
theory, and many other areas of mathematics. The best known fields are the field of
rational numbers, the field of real numbers and the field of complex numbers. Many other
fields, such as fields of rational functions, algebraic function fields, algebraic number
fields, and p-adic fields are commonly used and studied in mathematics, particularly in
number theory and algebraic geometry. Most cryptographic protocols rely on finite fields,
i.e., fields with finitely many elements.
9. MODULES
In mathematics, a module is a generalization of the notion of vector space in which the
field of scalars is replaced by a ring. The concept of module generalizes also the notion of
abelian group, since the abelian groups are exactly the modules over the ring of integers.
Like a vector space, a module is an additive abelian group, and scalar multiplication is
distributive over the operation of addition between elements of the ring or module and is
compatible with the ring multiplication.
10. VECTOR SPACES
A vector space or a linear space is a group of objects called vectors, added collectively
and multiplied (“scaled”) by numbers, called scalars. Scalars are usually considered to be
real numbers. But there are few cases of scalar multiplication by rational numbers,
complex numbers, etc. with vector spaces. The methods of vector addition and scalar
multiplication must satisfy specific requirements such as axioms. Real vector space and
complex vector space terms are used to define scalars as real or complex numbers.
11. LATTICES
A lattice is an abstract structure studied in the mathematical subdisciplines of order
theory and abstract algebra. It consists of a partially ordered set in which every pair of
elements has a unique supremum (also called a least upper bound or join) and a unique
infimum (also called a greatest lower bound or meet). An example is given by the power
set of a set, partially ordered by inclusion, for which the supremum is the union and the
infimum is the intersection. Another example is given by the natural numbers, partially
ordered by divisibility, for which the supremum is the least common multiple and the
infimum is the greatest common divisor. Lattices can also be characterized as algebraic
structures satisfying certain axiomatic identities. Since the two definitions are equivalent,
lattice theory draws on both order theory and universal algebra. Semilattices include
lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures
all admit order-theoretic as well as algebraic descriptions.
12. ALGEBRA OVER A FIELD
In mathematics, an algebra over a field (often simply called an algebra) is a vector space
equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a
set together with operations of multiplication and addition and scalar multiplication by
elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
