# Simplicity of rings

A ring $$R$$ is said to be simple if $$\lbrace 0 \rbrace$$ and $$R$$ are its only two-sided ideals. The ring $$R$$ is said to be left-semisimple (resp. right-semisimple) if the regular module $$_RR$$ (resp. $$R_R$$) is left-semisimple (resp. right-semisimple), which means that every submodule $$N$$ of $$_RR$$ (resp. $$R_R$$) is a direct summand. The simplicity of $$R$$ doesn't necessarily imply that $$R$$ is left or right semisimple. However if $$R$$ is a commutative simple ring (i.e. a field), then $$R$$ is both left and right semisimple since the ideals of $$R$$ correspond with the submodules of $$_RR$$ and $$R_R$$ and the only ideals of $$R$$ are $$\lbrace 0 \rbrace$$ and $$R$$.

My question is the following: is there a notion of left-simplicity (resp. right-simplicity), i.e. $$\lbrace 0 \rbrace$$ and $$R$$ are the only left (resp. right) ideals of $$R$$? I've done some reading and they never mentioned something alike.