Primitive ring

Remarks

Example: let $V$ be a vector space over a field $k$ and $R = \End_k(V)$. $V$ is a simple $R$-module, since any nonzero vector can be sent to any vector by a linear map. It is faithful, since a linear map that kills every vector is $0$. Non-example: $\Z$ is not primitive, since its simple modules are the groups $\Z/p\Z$, and $p$ kills $\Z/p\Z$; no simple module is faithful. A commutative ring is primitive if and only if it is a field. Every simple ring with identity is left and right primitive. A left Artinian ring is left primitive if and only if it is isomorphic to $M_n(D)$ for a division ring $D$ and an integer $n \ge 1$. There are left primitive rings that are not right primitive.