Maximal k-ideals and r-ideals in semirings

Abstract

Some properties of (left) k-ideals and r-ideals of a semiring are considered by the help of the congruence class semiring. It is proved that a proper k-ideal of a semiring with an identity is prime if it is a maximal left k-ideal. An equivalent condition for a proper r-ideal of a semiring being a maximal (left) r-ideal is established. It is shown that (left) r-ideals and (left) k-ideals coincide for an additively idempotent semiring, though the former is a special kind of the latter in general. It is proved that a proper k-ideal of an incline with an identity is a maximal k-ideal if and only if the corresponding congruence class semiring is the Boolean semiring.