The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

* Alternative algebra, an abstract algebra with alternative multiplication

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.

In abstract algebra, the derivative is interpreted as a morphism of modules of Kähler differentials.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form ( the latter only being possible when the dynamical system is linear ).

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

Elementary row operations are used to reduce a matrix to what is called triangular form ( in numerical analysis ) or row echelon form ( in abstract algebra ).

The group GL ( n, F ) and its subgroups are often called linear groups or matrix groups ( the abstract group GL ( V ) is a linear group but not a matrix group ).

The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space.

Compact matrix quantum groups are abstract structures on which the " continuous functions " on the structure are given by elements of a C *- algebra.

Most associative algebras considered in abstract algebra, for instance group algebras, polynomial algebras and matrix algebras, are unital, if rings are taken to be so.

represents the abstract matrix multiplication

ITL uses the abstract interface of matrix – vector, vector – vector, and vector – scalar operations MTL is default to serve those operations.

If abstract index notation is used also for spinors then these will carry a spinorial index and the Dirac gamma will carry one Lorentzian and two spinorian indices, but it is more common to regard spinors as column matrices and the Dirac gamma as a matrix which additionally carries a Lorentzian index.

Note that physicists often refer to this matrix or the coordinates themselves as the metric ( see, however, abstract index notation ).

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

In abstract algebra, a monoid ring is a new ring constructed from some other ring and a monoid.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

In abstract algebra, a congruence relation ( or simply congruence ) is an equivalence relation on an algebraic structure ( such as a group, ring, or vector space ) that is compatible with the structure.

From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.

In the branch of mathematics known as abstract algebra, a ring is an algebraic concept abstracting and generalizing the algebraic structure of the integers, specifically the two operations of addition and multiplication.

In abstract algebra, two nonzero elements and of a ring are respectively called a left zero divisor and a right zero divisor if ; this is a partial case of divisibility in rings.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.