Convex ideals of partially pseudo-ordered rings

Characteristics of partially pseudo-ordered ($K$-ordered) rings are considered. Properties of the set of all convex directed ideals in pseudo-ordered rings are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered rings the same role as convex directed subgroups for the theory of partially ordered groups. Necessary and sufficient conditions for a convex directed ideal of an $AO$-pseudo-ordered ring to be a rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an $AO$-pseudo-ordered ring form the root system for the lattice of all convex directed ideals of that ring. Properties of regular ideals for partially pseudo-ordered rings are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice pseudo-ordered rings.