Editing Simple ring (section)

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.