Maximal k-ideals and r-ideals in semirings

Abstract

Communicated by J. RosenthalSome 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.

$k$-Congruences and the Zariski topology in semirings

“…In this section, we recall some known definitions and facts [4][5][6][7]. Throughout this paper, R denotes a semiring (R, +, •), unless otherwise stated.…”

“…And Zhou and Yao [20] proved the existence of a one-to-one correspondence between the ideals which are lower sets and regular congruences in any additively idempotent semiring. By Han [6], the ideals which are lower sets are just k-ideals and every regular congruence is a k-congruence by Remark 3.9. Hence Theorem 3.8( 1) is a generalization of Theorem 3.8 in [10] and Theorem 5 in [20].…”

“…In order to narrow the gap, Henriksen [8] in 1958 defined k-ideals in semirings. From the fact that k-ideals coincide with ring ideals in any ring, k-ideals in semirings have been a very interesting subject of research [6,12,15,18,19].…”

k-Congruences on semirings

“…In order to narrow the gap, Henriksen [8] in 1958 defined k-ideals in semirings, which are a special kind of semiring ideals much closer to ring ideals than the general ones. Since then, many researchers have developed k-ideal theory [1,2,6,9,11,14,17,18].…”

“…It turns out that characteristic one semirings are additively idempotent commutative semirings and saturated ideals are just kideals. And it was proved in Han [6] that the ideals which are lower sets are nothing but k-ideals in additively idempotent semirings.…”