In ring theory and related areas of mathematics a central simple algebra (CSA) over K, also called a Brauer algebra after Richard Brauer, is a finite-dimensional (associative) algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large). In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Euclid, detail from The School of Athens by Raphael. ... Richard Dagobert Brauer (February 10, 1901 - April 17, 1977) was a leading German and American mathematician. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...

According to Artin–Wedderburn theorem a simple algebra A is isomorphic to M(n,S) for some skew field S. Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F , A and B are called similar if their skew fields S and T are isomorphic. The of set all equivalence classes of central simple algebras over a given field F can be equipped with a group operation and is called the Brauer group Br(F). In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. ... ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ...

Examples

• the quaternion and more generally any quaternion algebra

In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebra A over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. ...

Properties

• Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem-Noether theorem)
• If S is a simple subalgebra of a central simple algebra A then dim_FS divides dim_FA
• Every 4 dimensional central simple algebra over a field F is isomorpic to a quaternion algebra; in fact it is either a two-by-two matrix algebra, or a division algebra.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ... In mathematics, the Skolem-Noether theorem is a result on automorphisms of simple rings. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...

See also

• Brauer group