nLab
simple algebra

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

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

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

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

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

By the Artin–Wedderburn theorem, any finite-dimensional simple algebra over $k$ is a matrix algebra with entries lying in some division algebra whose center is $k$.