Abstract algebra

geometric object that has magnitude (or length) and direction

thumb|upright=1.2|From left to right: a square (geometry)|square, a [[cube and a tesseract. The square is two-dimensional (2D) and bounded by one-dimensional line segments; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes. ]] [[File:Dimension levels.svg|thumb|upright=1.2| The first four spatial dimensions, represented in a two-dimensional picture.

branch of mathematics studying algebraic structures and their relations

commutative ring in which every nonzero element is inversible

of a number x, 1 divided by x

mapping that preserves the operations of addition and scalar multiplication

vectors that map to their scalar multiples, and the associated scalars

set equipped with one or more finitary operations defined on it

sequence whose elements become arbitrarily close to each other

number that, when added to the original number, yields zero

property of a set of vectors of a vector space

formula that represents a mathematical object

thumb|200px|right|class=skin-invert-image|The transpose AT of a matrix A can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.

element with an inverse with respect to a given mathematical operation; element that can 'undo' the effect of another given element

In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.

the smallest superset of a given set that is closed under a given operation

thumb|right|400px|An w:Automorphism|automorphism of the Klein four-group shown as a mapping between two Cayley graphs, a permutation in cycle notation, and a mapping between two Cayley tables.

smallest vector subspace containing a given subspace

linear functional on tensor product square of a vector space

ring element such that can be multiplied by a non-zero element to equal 0

equivalence relation in algebra

n x n invertible matrices over a ring

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

generalization of a polynomial, where the number of terms is allowed to be infinite, defined algebraically without consideration of convergence (so that e.g. evaluation is not always defined)

irreducible element in the ring of polynomials; a non-constant polynomial that is not the product of two non-constant polynomials

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

an element which, when added to any element x in the set, yields x

complex square matrix such that its its conjugate transpose is equal to its negative

generalization of the Cartesian product

subgroup of a group G that each leaves invariant each element of a given subset of a G-set

concept used in algebra

linear algebra

colimit of a "directed family of objects"

construction to "glue together" mathematical objects along a downward-directed set

The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the corank of .

theorem

indexed set of subobjects of an algebraic structure

operation on elements of a polynomial ring which mimics the form of the derivative from calculus

operation in abstract algebra composing objects into "more complicated" objects

a “generic” algebraic structure over the given set, fulfilling no other equations except those given by the defining axioms

convex and balanced set

graph, constructed for a group

function which measures the "size" of elements in a field or integral domain

mathematical property shared by many number systems, that a product cannot be zero unless one of its factors is zero

In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an algebra over O to be a set together with concrete operations on this set that behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group

element of a generating set, a subset of an algebraic structure that allows specifying all elements of the structure

In algebra, the coimage of a homomorphism

basis of an algebraic structure that is canonical in a sense that depends on the precise context

algorithm in mathematics

In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory.