Editing Simple ring

In [[abstract algebra]], a branch of [[mathematics]], a '''simple ring''' is a [[zero ring|non-zero]] [[ring (mathematics)|ring]] that has no two-sided [[ideal (ring theory)|ideal]] besides the [[zero ideal]] and itself. In particular, a [[commutative ring]] is a simple ring if and only if it is a [[field (mathematics)|field]].

The [[Center (ring theory)|center]] of a simple ring is necessarily a field. It follows that a simple ring is an [[associative algebra]] over this field. It is then called a '''simple algebra''' over this field.

Several references (e.g., {{harvtxt|Lang|2002}} or {{harvtxt|Bourbaki|2012}}) require in addition that a simple ring be left or right [[artinian ring|Artinian]] (or equivalently [[semi-simple ring|semi-simple]]). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called '''quasi-simple'''.

Rings which are simple as rings but are not a [[simple module]] over themselves do exist: a full [[matrix ring]] over a field does not have any nontrivial two-sided ideals (since any ideal of <math>M_n(R)</math> is of the form <math>M_n(I)</math> with <math>I</math> an ideal of <math>R</math>), but it has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).

An immediate example of a simple ring is a [[division ring]], where every nonzero element has a multiplicative inverse, for instance, the [[quaternion|quaternions]]. Also, for any <math>n \ge 1</math>, the algebra of <math>n \times n</math> matrices with entries in a [[division ring]] is simple.

[[Joseph Wedderburn]] proved that if a ring <math>R</math> is a finite-dimensional simple algebra over a field <math>k</math>, it is isomorphic to a [[matrix algebra]] over some [[division algebra]] over <math>k</math>.  In particular, the only simple rings that are [[finite-dimensional algebra]]s over the [[real number]]s are rings of matrices over either the real numbers, the [[complex number]]s, or the [[quaternion]]s.

Wedderburn proved these results in 1907 in his doctoral thesis, ''On hypercomplex numbers'', which appeared in the [[Proceedings of the London Mathematical Society]]. His thesis classified finite-dimensional simple and also [[semisimple algebra]]s over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a Cartesian product, in the sense of algebras, of finite-dimensional simple algebras.

One must be careful of the terminology: not every simple ring is a [[Semisimple_module#Semisimple_rings|semisimple ring]], and not every simple algebra is a semisimple algebra.  However, every finite-dimensional simple algebra is a semisimple algebra, and every simple ring that is left- or right-[[Artinian ring|artinian]] is a semisimple ring.

Wedderburn's result was later generalized to [[semisimple ring]]s in the [[Wedderburn–Artin theorem]]: this says that every semisimple ring is a finite product of matrix rings over division rings.  As a consequence of this generalization, every simple ring that is left- or right-[[Artinian ring|artinian]] is a matrix ring over a division ring.

== 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.

== See also ==
* [[Simple (algebra)]]
* [[Simple algebra (universal algebra)]]

== References ==
* {{cite book | last1=Albert | first1=A. A. | author-link1=Abraham Adrian Albert | title=Structure of Algebras | series=Colloquium publications | volume=24 | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-1024-3 | p=37 }}
* {{citation | last1=Bourbaki | first1=Nicolas | author-link1=Nicolas Bourbaki | title=Algèbre Ch. 8 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-35315-7 | year=2012 }}
* {{cite journal | last1 = Nicholson | first1 = William K. | year = 1993 | title = A short proof of the Wedderburn-Artin theorem | journal = New Zealand J. Math | volume = 22 | pages = 83–86 |url = https://www.thebookshelf.auckland.ac.nz/docs/NZJMaths/nzjmaths022/nzjmaths022-01-010.pdf }}
* {{cite journal | last1 = Henderson | first1 = D. W. | year = 1965 | title = A short proof of Wedderburn's theorem | journal = Amer. Math. Monthly | volume = 72 | pages = 385–386 | doi=10.2307/2313499 }}
* {{citation | last1=Lam | first1=Tsit-Yuen | author-link1=Tsit Yuen Lam| title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 |mr=1838439 | year=2001 | doi=10.1007/978-1-4419-8616-0}}
* {{Citation | last1=Lang | first1=Serge | author-link1=Serge Lang |title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0387953854 |year=2002 }}
* {{citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Basic Algebra II | publisher=W. H. Freeman | edition=2nd | isbn=978-0-7167-1933-5 | year=1989 }}
[[Category:Ring theory]]