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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.

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You haven't heard about it because those conditions (either of them) are equivalent to being a division ring.

See this for example.

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