A unital associative algebra over a commutative ring is simple if it is it is a simple object in the category of --bimodules.
This can be stated in more elementary terms in any of the following equivalent ways:
- is nontrivial and has no nontrivial two-sided ideals.
- has exactly two two-sided ideals (which must be itself and its zero ideal).
In constructive algebra, this is too strong; we must say:
- For each two-sided ideal , is the zero ideal if and only if is proper (not equal to ).
By the Artin–Wedderburn theorem, any finite-dimensional simple algebra over is a matrix algebra with entries lying in some division algebra whose center is .
Created on July 18, 2010 11:17:07
by John Baez