Semiprime ring

Remarks

Equivalent forms: $R$ has no nonzero nilpotent ideal; for every two-sided ideal $I$, $I^2 = 0$ implies $I = 0$; the prime radical of $R$ is $0$. Example: $\Q \times \Q$ is semiprime: if $a = (a_1, a_2)$ and $aRa = 0$, then taking $r = 1$ gives $a_1^2 = a_2^2 = 0$, so $a = 0$. Non-example: in the ring $R$ of upper triangular $2 \times 2$ matrices over $\Q$, the matrix unit $a = e_{12}$ is nonzero and $e_{12} R e_{12} = 0$. Every prime ring is semiprime. A commutative ring is semiprime if and only if it is reduced.