mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible...

left-distributive and right-distributive, and thus distributive. Division is often shown in algebra and science by placing the dividend over the divisor with...

In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center...

normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified...

is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra. If A is a Euclidean Hurwitz algebra and a is in A, define the involution...

quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending...

mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over...

N(x)=xx^{*}} is called the norm of the algebra. A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of...

In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined...

central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and...

definition, which is more often encountered in modern textbooks. Using linear algebra, a projective space of dimension n is defined as the set of the vector...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x...

numbers C. The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic...

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented...

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative...

to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form...

finite-dimensional simple algebra over a field k {\displaystyle k} , it is isomorphic to a matrix algebra over some division algebra over k {\displaystyle...

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R {\displaystyle R} -algebras where R {\displaystyle R} need not...

idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table...

multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure...

Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems...

associative division algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the...

for algebras over a field k. If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra...

mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables...

of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation...

German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental...

composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with...