nLab
simple ring

A ring $R$ is simple if it is it is a simple object in the category of $R$-$R$-bimodules.

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

• $R$ is nontrivial and has no nontrivial two-sided ideals.
• $R$ has exactly two two-sided ideals (which must be $R$ 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 $R$).