Geodesic
$S$−$k$−prime and $S$−$k$−semiprime ideals of semirings
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 2, pp. 301-317.

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Let R be a commutative ring and S a multiplicatively closed subset of R. Hamed and Malek defined an ideal P of R disjoint with S to be an S-prime ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ P, then sa ∈ P or sb ∈ P. In this paper, we introduce the notions of S-k-prime and S-k-semiprime ideals of semirings, S-k-m-system, and S-k-p-system. We study some properties and characterizations for S-k-prime and S-k-semiprime ideals of semirings in terms of S-k-m-system and S-k-p-system respectively. We also introduce the concepts of S-prime semiring and S-semiprime semiring and study the characterizations for S-k-prime and S-k-semiprime ideals in these two semirings.
Keywords: Semiring, $S$−$k$prime ideal, $S$−$k$−semiprime ideal, $S$−prime semiring, $S$−semiprime semiring 
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Bhowmick, Sumon; Goswami, Jituparna; Kar, Sukhendu. $S$−$k$−prime and $S$−$k$−semiprime ideals of semirings. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 2, pp. 301-317. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_2_a4/

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