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

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...

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

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

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...

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...

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...

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...

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

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

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...

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras...

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...

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...

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...

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, 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...

b, c such that if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions of isotopy are equivalent...

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

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

spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) to physics. Spacetime algebra provides...

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...

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...

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

= 1. Banach algebra Composition algebra Division algebra Gelfand–Mazur theorem Hurwitz's theorem (composition algebras) "Normed Algebra". Encyclopaedia...

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

universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...