1
$\begingroup$

I'm reading Clausen, Fast Fourier Transforms, and he states (p. 34):

Let A be an associative algebra.
A is semisimple $\Leftrightarrow$ A is a direct sum of minimal left ideals.

Can anyone explain why?

Followup question: since A is semisimple, its left A-modules are completely reducible, i.e. every left A-module is the direct sum of simple left A-modules. Is there an easy correspondence between the simple A-modules and the left ideals of A?

  • 0
    http://www.math.uni-duesseldorf.de/~wisbauer/book.pdf2012-11-23

1 Answers 1