Simple ring

Remarks

Example: let $F$ be a field and $R = M_2(F)$. Let $I$ be a nonzero two-sided ideal and $A \in I$ with an entry $a_{ij} \ne 0$. With the matrix units $E_{kl}$, one has $E_{ki} A E_{jl} = a_{ij} E_{kl}$, so every $E_{kl}$ lies in $I$, hence $I = R$.

Non-example: $\Z$ is not simple, since $2\Z$ is a two-sided ideal different from $0$ and $\Z$.

A commutative ring is simple if and only if it is a field. A simple ring need not be Artinian: the Weyl algebra over a field of characteristic $0$ is simple and has no zero divisors, but it is not a division ring. The center of a simple ring is a field. Some authors require a simple ring to be Artinian as well; here no such condition is imposed.