nLab
semisimple algebra

Context

Algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (∞,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Contents

Definition

An associative unital algebra AA over a field kk is semisimple if its Jacobson radical is trivial.

Properties

If AA is finite-dimensional, this is equivalent to saying that AA is a finite product of finite-dimensional simple algebras.

By the Artin-Wedderburn theorem, any finite-dimensional simple algebra over kk is a matrix algebra with entries lying in some division algebra whose center is kk. So, every finite-dimensional semisimple algebra is a finite product of such matrix algebras.

Last revised on October 14, 2012 at 16:50:26. See the history of this page for a list of all contributions to it.