SIMPLE RING

In abstract algebra, a 'simple ring' is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra.
According to the Artin-Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.
Any quotient of a ring by a maximal ideal is a simple ring. A ring ''R'' is simple if and only its opposite ring ''R''^o is simple.
An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

Contents

Wedderburn's theorem

See also

Reference

Wedderburn's theorem

Wedderburn's theorem characterizes simple rings with a unit and a minimal left ideal. (The left Artinian condition is a generalization of the second assumption.) Namely it says that every such ring is, up to isomorphism, a ring of ''n'' × ''n'' matrices over a division ring.
Let ''D'' be a division ring and M(n,''D'') be the ring of matrices with entries in ''D''. It is not hard to show that every left ideal in M(n,''D'') takes the following form:
:{''M'' ∈ M(n,''D'') | The ''n''₁...''n_k''-th columns of ''M'' have zero entries},
for some fixed {''n''₁,...,''n_k''} ⊂ {1, ..., ''n''}. So a minimal ideal in M(n,''D'') is of the form
:{''M'' ∈ M(n,''D'') | All but the ''k''-th columns have zero entries},
for a given ''k''. In other words, if ''I'' is a minimal left ideal, then ''I'' = (M(n,''D'')) ''e'' where ''e'' is the idempotent matrix with 1 in the (''k'', ''k'') entry and zero elsewhere. Also, ''D'' is isomorphic to ''e''(M(n,''D''))''e''. The left ideal ''I'' can be viewed as a right-module over ''e''(M(n,''D''))''e'', and the ring M(n,''D'') is clearly isomorphic to the algebra of homorphisms on this module.
The above example suggests the following lemma:

'Lemma.' ''A'' is a ring with identity 1 and an idempotent element ''e'' where ''AeA = A''. Let ''I'' be the left ideal ''Ae'', considered as a right module over ''eAe''. Then ''A'' is isomorphic to the algebra of homomorphisms on ''I'', denoted by ''Hom''(''I'').

'Proof:' We define the "left regular representation" Φ : ''A'' → ''Hom''(''I'') by Φ(''a'')''m'' = ''am'' for ''m'' ∈ ''I''. Φ is injective because if ''a · I'' = ''aAe'' = 0, then ''aA'' = ''aAeA'' = 0, which implies ''a'' = ''a'' · 1 = 0.
For surjectivity, let ''T'' ∈ ''Hom''(''I''). Since ''AeA'' = ''A'', the unit 1 can be expresses as 1 = ∑''a_ieb_i''. So
:''T''(''m'') = ''T''(1·''m'') = ''T''(∑''a_ieb_im'') = ∑ ''T''(''a_ieeb_im'') = ∑ ''T''(''a_ie) eb_im'' = [ ∑''T''(''a_ie'')''eb_i'']''m''.
Since the expression [∑''T''(''a_ie'')''eb_i''] does not depend on ''m'', Φ is surjective. This proves the lemma.

Wedderburn's theorem follows readily from the lemma.

'Theorem' ('Wedderburn'). If ''A'' is a simple ring with unit 1 and a minimal left ideal ''I'', then ''A'' is isomorphic to the ring of ''n'' × ''n'' matrices over a division ring.

One simply has to verify the assumptions of the lemma hold, i.e. find an idempotent ''e'' such that ''I = Ae'' and ''A = AeA''.
''A'' being simple also implies ''eAe'' is a division ring.

See also

★ simple (algebra)

Reference

★ D.W. Henderson, A short proof of Wedderburn's theorem, ''Amer.'' ''Math.'' ''Monthly'' '72' (1965), 385-386.