Simple algebra - Citizendia

In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set {ab | a, b are elements of the algebra} ≠ {0}. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

The second condition in the definition precludes the following situation: consider the algebra

$\{ \begin{bmatrix} 0 & \alpha \\ 0 & 0 \\ \end{bmatrix}\, | \, \alpha \in \mathbb{C} \}$

with the usual matrix operations. This is a one-dimensional algebra in which the product of any two elements is zero. This condition ensures that the algebra has a minimal nonzero left ideal, which simplifies certain arguments.

An immediate example of simple algebras are division algebras, where every element has a multiplicative inverse, for instance, the real algebra of quaternions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Also, one can show that the algebra of n × n matrices with entries in a division ring is simple. In fact, this characterizes all simple algebras up to isomorphism, i. e. any simple algebra is isomorphic to a matrix algebra over some division ring. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible This result was given in 1907 by Joseph Wedderburn in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. Joseph Henry Maclagan Wedderburn ( 2 February 1882 Forfar Angus, Scotland – 9 October 1948, Princeton New Jersey A dissertation (also called thesis or disquisition) is a document that presents the author's Research and findings and is submitted in support of candidature The London Mathematical Society ( LMS) is the leading mathematical society in England. Wedderburn's thesis classified simple and semisimple algebras. In Ring theory, a semisimple algebra is an Associative algebra which has trivial Jacobson radical (that is only the zero element of the algebra is in the Simple algebras are building blocks of semi-simple algebras: any (finite dimensional) semi-simple algebra is a Cartesian product, in the sense of algebras, of simple algebras.

Wedderburn's result was later generalized to semisimple rings in the Artin–Wedderburn theorem. In Mathematics, especially in the area of Abstract algebra known as Module theory, a semisimple module or completely reducible module is a type In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings.

Examples

A central simple algebra (sometimes called Brauer algebra) is a simple finite dimensional algebra over a field F whose center is F. In Ring theory and related areas of Mathematics a central simple algebra ( CSA) over a field K (also called a Brauer algebra In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements

Simple universal algebras

In universal algebra, an abstract algebra A is called "simple" if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism with domain A is either injective or constant. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" ↔ See Congruence (geometry for the term as used in elementary geometry

As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra.

Examples

Simple universal algebras

See also