Remark 2

We sometimes say that a ring A is quasi-simple if it is not reduced to 0 and if its only two-sided ideals are 0 and A. We say that A is primitive if it admits a faithful simple module. By Proposition 1, every simple ring is quasi-simple. Since every nonzero ring admits a simple module and the annihilator of a simple module is a two-sided ideal, we see that every quasi-simple ring is primitive. However, there exist quasi-simple rings that are not simple and primitive rings that are not quasi-simple (VIII, p. 128, Exercise 2); such rings are not left Artinian.