Prime radicals of directed pseudo-ordered rings

Characteristics of partially pseudo-ordered ($\mathcal{K}$-ordered) rings are considered. Properties of ideals in pseudo-ordered rings are described. A variation of the concept of the prime radical for a ring in the subclass of directed pseudo-ordered rings ($\mathcal{AO}$-ordered rings) is investigated. The description of elements of $\mathcal{AO}$-prime radicals for $\mathcal{AO}$-ordered rings is obtained.