nLab simple algebra

A unital associative algebra AA over a commutative ring kk is simple if it is it is a simple object in the category of AA-AA-bimodules.

This can be stated in more elementary terms in any of the following equivalent ways:

  • AA is nontrivial and has no nontrivial two-sided ideals.
  • AA has exactly two two-sided ideals (which must be AA itself and its zero ideal).

In constructive algebra, this is too strong; we must say:

  • For each two-sided ideal II, II is the zero ideal if and only if II is proper (not equal to AA).

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.

Created on July 18, 2010 at 11:10:07. See the history of this page for a list of all contributions to it.