Simple ring

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a simple ring is a <a href="/facts/Zero_ring/trFjtDJ1">non-zero</a> <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> that has no two-sided <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> besides the <a href="/facts/Zero_ideal/PvH2qhxh">zero ideal</a> and itself. In particular, a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> is a simple ring if and only if it is a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>.
The <a href="/facts/Center_(ring_theory)/kJcejgkj">center</a> of a simple ring is necessarily a field. It follows that a simple ring is an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over this field. It is then called a simple algebra over this field.
Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right <a href="/facts/Artinian_ring/BlzWdfTL">Artinian</a> (or equivalently <a href="/facts/Semi-simple_ring/JIXNLzck">semi-simple</a>). 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 <a href="/facts/Simple_module/00roKQrG">simple module</a> over themselves do exist: a full <a href="/facts/Matrix_ring/oj5HBr8B">matrix ring</a> over a field does not have any nontrivial two-sided ideals (since any ideal of 
 
 
 
 
 M
 
 n
 
 
 (
 R
 )
 
 
 {\displaystyle M_{n}(R)}
 
 is of the form 
 
 
 
 
 M
 
 n
 
 
 (
 I
 )
 
 
 {\displaystyle M_{n}(I)}
 
 with 
 
 
 
 I
 
 
 {\displaystyle I}
 
 an ideal of 
 
 
 
 R
 
 
 {\displaystyle R}
 
), 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 <a href="/facts/Division_ring/bCf07uRj">division ring</a>, where every nonzero element has a multiplicative inverse, for instance, the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>. Also, for any 
 
 
 
 n
 ≥
 1
 
 
 {\displaystyle n\geq 1}
 
, the algebra of 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 matrices with entries in a <a href="/facts/Division_ring/bCf07uRj">division ring</a> is simple.
<a href="/facts/Joseph_Wedderburn/HEkrKh00">Joseph Wedderburn</a> proved that if a ring 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is a finite-dimensional simple algebra over a field 
 
 
 
 k
 
 
 {\displaystyle k}
 
, it is isomorphic to a <a href="/facts/Matrix_algebra/oj5HBr8B">matrix algebra</a> over some <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a> over 
 
 
 
 k
 
 
 {\displaystyle k}
 
. In particular, the only simple rings that are finite-dimensional algebras over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> are rings of matrices over either the real numbers, the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, or the <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>.
Wedderburn proved these results in 1907 in his doctoral thesis, On hypercomplex numbers, which appeared in the <a href="/facts/Proceedings_of_the_London_Mathematical_Society/jdcf3vhX">Proceedings of the London Mathematical Society</a>. His thesis classified finite-dimensional simple and also <a href="/facts/Semisimple_algebra/JIXNLzck">semisimple algebras</a> 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 <a href="/facts/Semisimple_module/9KASYk35">semisimple ring</a>, 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-<a href="/facts/Artinian_ring/BlzWdfTL">artinian</a> is a semisimple ring.
Wedderburn's result was later generalized to <a href="/facts/Semisimple_ring/9KASYk35">semisimple rings</a> in the <a href="/facts/Wedderburn%25E2%2580%2593Artin_theorem/EB5rY5sd">Wedderburn–Artin theorem</a>: 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-<a href="/facts/Artinian_ring/BlzWdfTL">artinian</a> is a matrix ring over a division ring.

Simple ring open-in-new

Simple ring