High Quality Content by WIKIPEDIA articles! In abstract algebra, a simple ring is a non-zero ring that has no (two-sided) ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left (or right) module over itself (although both notions coincide in the commutative setting). Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of M(n,R) is of the form M(n,I) with I and ideal of R), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns).