Editing Simple ring (section)

== Examples ==
Let <math>\mathbb{R}</math> be the field of real numbers, <math>\mathbb{C}</math> be the field of complex numbers, and <math>\mathbb{H}</math> the [[quaternion]]s.
* A [[central simple algebra]] (sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a [[field (mathematics)|field]] <math>F</math> whose [[center of an algebra|center]] is <math>F</math>.
* Every finite-dimensional simple algebra over <math>\mathbb{R}</math> is isomorphic to an algebra of <math>n \times n</math> matrices with entries in <math>\mathbb{R}</math>, <math>\mathbb{C}</math>, or <math>\mathbb{H}</math>.  Every [[central simple algebra]] over <math>\mathbb{R}</math> is isomorphic to an algebra of <math>n \times n</math> matrices with entries <math>\mathbb{R}</math> or <math>\mathbb{H}</math>.  These results follow from the [[Frobenius theorem (real division algebras)|Frobenius theorem]].
* Every finite-dimensional simple algebra over <math>\mathbb{C}</math> is a central simple algebra, and is isomorphic to a matrix ring over <math>\mathbb{C}</math>.
* Every finite-dimensional central simple algebra over a [[finite field]] is isomorphic to a matrix ring over that field.
* Over a field of characteristic zero, the [[Weyl algebra]] is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply to it.