Prime ideal in a noncommutative ring

Summary

A prime ideal of a noncommutative ring is a proper two-sided ideal $P$ with the following property. A product $AB$ of two-sided ideals lies in $P$ only when $A$ or $B$ does. For commutative rings this is the usual prime ideal. Ideals are used instead of elements because, for $n \ge 2$, the matrix ring $M_n(k)$ over a field has no proper ideal with the element-wise property. Prime ideals define the prime radical and are the kernels of maps onto prime rings.