Concept
Algebra over a field
Related concepts (49)
Algebra
Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures. "Subalgebra" can refer to either case. A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors.
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Module (mathematics)
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.
Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.
Power associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element is performed an operation by itself several times, it doesn't matter in which order the operations are carried out, so for instance .
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Abstract algebra
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.
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy This ensures that the multiplication operation is continuous. A Banach algebra is called unital if it has an identity element for the multiplication whose norm is and commutative if its multiplication is commutative.
Algebraic structure
In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures.