Introduction to Abstract Algebra

The Abstract algebra is a subject area of mathematics which studies algebraic structures such as rings, groups, vector spaces, modules, fields and algebra. The word "ABSTRACT ALGEBRA" was introduced at the turn of 20th century to distinguish this area from what was generally termed as algebra. The study of rules for manipulating formulae and algebraic expressions including unknown and real or complex numbers frequently known as elementary algebra.

Group Theory

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of group is vital to abstract algebra: the other well-known algebraic structures, like fields, rings, and vector spaces can all be seen as groups endowed with additional axioms and operations. The groups recur via mathematics and the techniques of group theory have very strongly influenced in many sections of algebra. The two sub-categories of group theory are linear algebraic groups and Lie groups which have experienced great advances and have become subject regions in their own right. 

Main classes of groups:

Permutation Groups:

Permutation group was the initial class of groups to undergo a organized study. Given any set S and a collection G of bisections of S into itself termed as permutations which is closed under compositions and inverses, here G is a group acting on S. When S consists of n elements and G consists of all permutations, then G is the symmetric group i.e. SG; in common, any permutation group G is a subgroup of the symmetric group of S. An early construction due to Cayley displayed any group as a permutation group, acting on itself (S = G) by means of the left regular representation.

In most situations, according to the properties of permutation group's action on the corresponding set of structure where it can be studied. For illustration, in this manner one proves that for n ≥ 5, the alternating group A_n is simple, which does not admit any appropriate normal subgroups. This fact plays a significant role in the impossibility of solving a normal algebraic equation of degree n ≥ 5 in radicals.

Matrix Groups:

The next class of groups is matrix groups or linear groups. Here S is a set having invertible matrices of given order n over a field K which is closed under the products and inverses. This kind of group acts on the n-dimensional vector space K_n by linear transformations. This kind of action makes matrix groups conceptually similar to permutation groups and the geometry of the action might be usefully exploited to establish the properties of group S.

Transformation Groups:

The special case of transformation groups are Permutation groups and matrix groups: the groups which act on a certain space G preserving its inherent structure. In case of permutation groups G is a set; and for matrix groups G is a vector space. The idea of a symmetry group is closely associated with the concept of a transformation group: transformation groups generally consist of all transformations which preserve a certain structure.

The hypothesis of transformation groups forms a bridge that connects group theory with differential geometry.

Abstract Groups:

Most of groups considered in the initial stage of the development of group theory were "concrete", having been realized via numbers, matrices and permutations. It was not present until the late nineteenth century that the idea of an abstract group as a set with operations fulfilling a certain system of axioms began to take hold. A common way of identifying an abstract group is through a presentation by generators and relations which is as follows:

An important source of abstract groups is given by the building of a factor group or quotient group, F/J of a group F by a common subgroup J. The class groups of algebraic number fields were among the initial cases of factor groups that have much interest in the number theory. When a group F is a permutation group on a set X, the factor group F/J is no longer performing on X; however the idea of an abstract group allows, not anyone to worry about this discrepancy.

The change of perspective from concrete to abstract groups makes it natural to consider the properties of groups which are independent of a specific realization or in modern language, invariant under isomorphism and also the classes of group with such a given property: that is the simple groups, finite groups, periodic groups, solvable groups and so forth. In place of exploring properties of a separate group one seeks to establish outcomes that can be applied to entire class of groups. The new concept was of paramount significance for the development of mathematics: it foreshadowed the formation of abstract algebra in the works of Emmy Noether, Hilbert, and mathematicians of their school.

Topological and Algebraic Groups:

A significant elaboration of the concept of a group takes place if G is endowed with additional structure, mainly, of a topological space, algebraic range or differentiable manifold. The group operations i (inversion) and m (multiplication) can be shown as follows:

Ring Theory:

Ring theory is the study of rings in abstract algebra-algebraic structures in which multiplication and addition are defined and have similar properties to those familiar from integers. Ring theory is the study of rings, their representations or in various languages, modules, special classes of rings (i.e., division rings, group rings, and universal enveloping algebras) and also an array of properties which proved to be of interest both in the theory itself and for its applications, like homological properties and polynomial identities.

The commutative rings are easily understood than non-commutative ones. The algebraic number theory and Algebraic geometry, that give many usual illustrations of commutative rings, that have driven much of the development of commutative ring theory, that is now under the commutative algebra, a key region of modern mathematics. Since of these three fields are so intimately connected it is normally difficult and meaningless to decide to which field belongs a specific outcome. For illustration, Hilbert's Nullstellensatz is a theorem which is a primary for algebraic geometry and is stated and proved in terms of commutative algebra. Likewise, Fermat's last theorem is stated in term of elementary arithmetic which is a section of commutative algebra but its proof includes deep answers of both algebraic number theory and algebraic geometry.

Non-commutative rings are slightly dissimilar in flavor, as more unusual behavior can arise. Whereas the theory has developed in its own right, a fairly present trend has sought to parallel the commutative development by building the theory of many classes of non-commutative rings in a geometric fashion as if they were rings of functions on 'non-commutative spaces'. This trend begins in the year 1980 with the development of non-commutative geometry and with the discovery of quantum groups. It leads to a enhanced understanding of non-commutative rings.

Field Theory:

The field is a mathematical unit for which addition, subtraction, multiplication and division are very well defined and the Field theory is a category of mathematics that studies the properties of fields.

Fields are main objects of study in algebra as they provide a helpful generalization of many number systems, such as real numbers, rational numbers, and complex numbers. Principally, the normal rules of associativity, distributivity, and commutativity hold. Fields also look in many other regions of mathematics.

Whenever abstract algebra was initially being developed, the definition of a field normally did not include the commutativity of multiplication and what we today call a field would have been called either a rational domain or a commutative field. In contemporary usage, the field is always commutative. A division ring or division algebra or at times a skew field is a structure that satisfies all the properties of a field except probably for commutativity. Non-commutative field is still broadly used. In French, fields are known as corps (literally, body), normally regardless of their commutative. Whenever essential, a commutative field is called and corps commutate when a skew field corps gauche. The term Korper is used to indicate field which is a German word, therefore the use of the blackboard bold is used to denote a field.

The concept of fields was initially used to prove that there is no common formula expressing in terms of radicals that is the roots of a polynomial with rational coefficients of degree 5 or higher.

Applications of Field Theory:

The concept of a field is of much use, for illustration, in defining matrices and vectors, the two structures in linear algebra whose components can be elements of a random field. In number theory the finite fields are used.  The Coding theory and Galois Theory and algebraic extension are significant tool. The binary fields and fields of characteristic are very useful in computer science.

Module Theory:

In abstract algebra, the concept of module over a ring is a generalization of the notion of vector space, in which the equivalent scalars are permitted to lie in a random ring. Modules also generalize the concept of abelian groups; such groups are modules over the ring of integers. Therefore, a module, like a vector space is an additive abelian group.  A product that is defined between elements of the ring and elements of the module is distributive over both the parameters and is compatible with ring multiplication. Modules are comparatively very close to the representation theory of groups.

Types of Modules:

Finitely generated - An R-module N is finitely created if there exists finitely many elements x1... xn in N such that each element of N is a linear combination of those elements with coefficients from the ring R.

Cyclic module -When a module is created by one element it is termed as cyclic module.

Free - A module that has a basis, or equivalently, one which is isomorphic to a direct sum of copies of the ring R is termed as free module. These are modules which behave likewise as vector spaces.

Projective - They are direct summands of the free modules and share most of their desirable properties.

Injective - Injective modules are explained dually to the projective modules.

Flat - A module is known as flat if taking the tensor outcome of it with any short exact series of R-modules which preserves the exactness.

Simple - A simple module S is a module which is not {0} and whose only sub modules are {0} and S. The simple modules are sometimes known as irreducible.

Semi-simple -A semi-simple module is a direct sum (finite or not) of all simple modules. These modules are also termed as completely reducible.

Indecomposable - An indecomposable module is a non-zero module which cannot be written as a direct sum of two non-zero sub modules. All simple modules are indecomposable; however there are few indecomposable modules that are not simple (example, uniform modules).

Faithful - A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. rx ≠ 0 for some x in M). Equally, the annihilator of M is the zero ideal.

Noetherian - A Noetherian module is a module which satisfies the ascending chain condition on sub modules, i.e. every increasing chain of sub modules becomes stationary after finitely many steps. Equally, every sub module is finitely generated.

Artinian - An Artinian module is a module which satisfies the descending chain condition on sub modules, i.e. every decreasing chain of sub modules becomes stationary after finitely many steps.

Graded - A graded module is a module with a decomposition as a direct sum M = ⊕x Mx over a graded ring R = ⊕x Rx such that RxMy ⊂ Mx + y for all x and y.

Uniform - A uniform module is a module in which all pairs of nonzero sub modules have nonzero intersection.

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