# Wedderburn-Artin theorem

From Encyclopedia of Mathematics

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A theorem which fully describes the structure of associative Artinian rings without nilpotent ideals. An associative ring $R$ has the minimum condition for right ideals and has no nilpotent ideals if and only if $R$ is the direct sum of a finite number of ideals, each of which is isomorphic to a complete matrix ring of finite order over a suitable skew-field; this decomposition into a direct sum is unique apart from the ordering of its terms. This theorem was first obtained by J. Wedderburn for finite-dimensional algebras over a field, and was proved by E. Artin [1] in its final formulation.

#### References

[1] | E. Artin, "The influence of J.H.M. Wedderburn on the development of modern algebra" Bull. Amer. Math. Soc. , 56 (1950) pp. 65–72 |

#### Comments

#### References

[a1] | J.H.M. Wedderburn, "Lectures on matrices" , Dover, reprint (1964) |

[a2] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) pp. 380, 369 |

[a3] | P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. 174ff |

**How to Cite This Entry:**

Wedderburn–Artin theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Wedderburn%E2%80%93Artin_theorem&oldid=23126