Simple ring

A simple ring is a ring $R$ such that its only two-sided ideals are $0$ and $R$ itself.

Simple rings are the “atoms” of ring theory: if $I$ is a nonzero two-sided ideal, then $R/I$ is a nontrivial quotient ring , so simplicity rules out all proper quotients. Standard families of examples are division rings and matrix rings over them.

Examples:

• If $D$ is a division ring, then $D$ is simple.
• For a division ring $D$ and $n\ge 1$, the ring $M_n(D)$ is simple.
• $\mathbb{Z}$ is not simple since $(2)$ is a nontrivial two-sided ideal.