Research Article ϕ -δ-Primary Hyperideals in Krasner Hyperrings Hao Guan , 1,2 Elif Kaya , 3 Melis Bolat , 4 Serkan Onar , 5 Bayram Ali Ersoy , 4 and Kostaq Hila 6 1 Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China 2 School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun, Guizhou 558000, China 3 Department of Mathematics and Science Education, Istanbul Sabahattin Zaim University, Istanbul, Turkey 4 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey 5 Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey 6 Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana, Albania Correspondence should be addressed to Kostaq Hila; kostaq_hila@yahoo.com Received 26 May 2022; Accepted 16 July 2022; Published 21 September 2022 Academic Editor: Hasan Dinçer Copyright © 2022 Hao Guan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we study commutative Krasner hyperrings with nonzero identity. ϕ-prime, ϕ-primary and ϕ-δ-primary hyperideals are introduced. e concept of δ-primary hyperideals is extended to ϕ-δ-primary hyperideals. Some characterizations of hyperideals are provided to classify them. e relation between ϕ-δ-primary hyperideals and other hyperideals is discussed. 1. Introduction In commutative ring theory, prime and primary ideals have a significant place. e importance of prime ideals encourages researchers to expand these concepts and find applications. Many different types of generalizations have been investi- gated by several authors, some of them [1–5]. Prime and primary ideals are generalized to ϕ -prime and ϕ-primary ideals. Let (R, +,.) be a commutative ring with nonzero identity. Denote the set of all ideals of R by L(R) (proper ideals of R by L ∗ (R). Let ϕ be a function such that ϕ: L(R) ⟶ L(R) ∪∅ { }. Let N be a proper ideal of R. N is called a ϕ-prime ideal [1], if ab ∈ N − ϕ(N), then either a ∈ N or b ∈ N for some a, b ∈ R. By the way, N is called a ϕ-primary ideal when, if ab ∈ N − ϕ(N), then a ∈ N or b k ∈ N for some a, b ∈ R,k ∈ N [3, 4]. e image of ideal, ϕ(N), can be equal 0, ∅, N, N 2 , N n , N w (w denotes the intersection of ideals of N i ). A proper ideal N of R is called weakly prime (primary) ideal respectively in [6] ([7]) if 0 ≠ ab ∈ N, for some a, b ∈ R, then a ∈ N or b ∈ N (b k ∈ N for some k ∈ N). Anderson generalized it in [1], where N is weakly ϕ-prime ideal when ϕ(N) 0. Zhao [8] introduced δ-primary ideal as an expansion of an ideal, δ: L(R) ⟶ L(R) is a function that meets the following requirements: i)N⊆δ(N), for all ideals N of R, ii) If N⊆M, where N and M are ideals of R, then δ(N)⊆δ(M), iii)δ(K ∩ L) δ(K) ∩ δ(L) for all ideals K, L of R. Entire of the δ ideal expansions provides the property δ 2  δ, which is δ(δ(N))  δ(N) for all ideal N of R [8]. A. Jaber chose ϕ such a reduction function in [9], which satisfies the following requirements: i) ϕ(N)⊆N, for all ideals N of R, ii) If N⊆M, where N and M are ideals of R, then ϕ(N)⊆ϕ(M). He obtained generalization of ϕ-δ-primary ideal by combining these two concepts. Let N be an ideal of R, δ be an ideal expansion and ϕ be an ideal reduction [9]. N is called ϕ- δ-primary if ab ∈ N − ϕ(N), then either a ∈ N or b ∈ δ(N), for all a, b ∈ R. Some results on φ − δ-primary ideals can be found in [10, 11]. e theory of hyperstructures was innovated by Marty in 1934 [12]. He defined hypergroupoid (G, ° ) for G ≠∅, P ∗ (G) represents family of nonempty subsets of G and ° :G × G ⟶ P ∗ (G) is a binary hyperoperation. Let (G, ° ) be a hypergroupoid. G is a semihypergroup, if ∀a, b, c ∈ G, a ° (b ° c)(a ° b) ° c, which means ∪ u∈a° b u ° c  ∪ v∈b° c a ° v. If Hindawi Mathematical Problems in Engineering Volume 2022, Article ID 1192684, 12 pages https://doi.org/10.1155/2022/1192684