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Wedderburn-Artin theorem (Theorem)

If $ R$ is a left semisimple ring, then

$\displaystyle R \cong \mathbb{M}_{n_1}(D_1) \times \cdot\cdot\cdot \times \mathbb{M}_{n_r}(D_r)$
where each $ D_i$ is a division ring and $ \mathbb{M}_{n_i}(D_i)$ is the matrix ring over $ D_i$, $ i = 1, 2, \ldots, r$. The positive integer $ r$ is unique, and so are the division rings (up to permutation).

Some immediate consequences of this theorem:

This theorem is a special case of the more general structure theorem on semiprimitive rings.



"Wedderburn-Artin theorem" is owned by CWoo. [ full author list (2) ]
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See Also: semiprimitive ring

Other names:  structure theorem on semisimple rings, Artin-Wedderburn theorem
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Cross-references: semiprimitive rings, fields, direct product, finite, commutative, isomorphic, ring, artinian, consequences, permutation, integer, positive, matrix ring, division ring, semisimple ring
There are 9 references to this entry.

This is version 11 of Wedderburn-Artin theorem, born on 2004-04-20, modified 2008-05-24.
Object id is 5785, canonical name is WedderburnArtinTheorem.
Accessed 4199 times total.

Classification:
AMS MSC: 16D70 (Associative rings and algebras :: Modules, bimodules and ideals :: Structure and classification , direct sum decomposition, cancellation)
Pending Errata and Addenda
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