Wedderburn-Artin theorem
If R is a left semisimple ring, then
R≅𝕄n1(D1)×⋯×𝕄nr(Dr) |
where each Di is a division ring and 𝕄ni(Di) is the matrix ring over Di, i=1,2,…,r. The positive integer r is unique, and so are the division rings (up to permutation).
Some immediate consequences of this theorem:
-
•
A simple (http://planetmath.org/SimpleRing) Artinian ring is isomorphic
to a matrix ring over a division ring.
-
•
A commutative
semisimple ring is a finite direct product
of fields.
This theorem is a special case of the more general theorem on semiprimitive rings.
Title | Wedderburn-Artin theorem |
---|---|
Canonical name | WedderburnArtinTheorem |
Date of creation | 2013-03-22 14:19:08 |
Last modified on | 2013-03-22 14:19:08 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 14 |
Author | CWoo (3771) |
Entry type | Theorem |
Classification | msc 16D70 |
Synonym | structure theorem on semisimple rings |
Synonym | Artin-Wedderburn theorem |
Related topic | SemiprimitiveRing |