In the book I've been reading recently (Algebra by Thomas W. Hungerford) the following definition is given:
A left module $A$ over a ring $R$ is simple provided that $RA\neq 0$ and $A$ has no proper submodules. A ring $R$ is simple if $R^2\neq 0$ and $R$ has no proper two-sided ideals.
My question is why don't we define a simple ring to be a ring which is simple as a module over itself (as in many definitions for rings that arise from definitions for modules). Clearly a ring which is simple as a module over itself is also simple, however the converse is not true (take the ring of square matrices over a simple ring).