IHJPAS. 36(2)2023 383 This work is licensed under a Creative Commons Attribution 4.0 International License. Abstract. A class of hyperrings known as divisible hyperrings will be studied in this paper. It will be presented as each element in this hyperring is a divisible element. Also shows the relationship between the Jacobsen Radical, and the set of invertible elements and gets some results, and linked these results with the divisible hyperring. After going through the concept of divisible hypermodule that presented 2017, later in 2022, the concept of the divisible hyperring will be related to the concept of division hyperring, where each division hyperring is divisible and the converse is achieved under conditions that will be explained in the theorem 3.14. At the end of this paper, it will be clear that the goal of this paper is to study the concept of divisible hyperring by giving some examples, remarks, and results that are related to the concept of divisible hyperrings. Keywords: divisible hyperring, divisible hypermodule, division hyperring, Jacobson radical. 1.Introduction. Based on the concept of divisible hypermodule that Sopon Boriboon and Sajee Pianskool introduced in their paper “Baer hypermodule over Krasner hyperring” [1] in 2017, and later by Hashem Bordbar and Irina Cristea in 2022 in the paper “Divisible hypermodule” [2]. This paper presents a study on the concept of divisible hyperring. Before that, would like to re-examine the concept of hyperstructure that was first introduced by the French mathematician Marty in 1937 in the following form. The function ⊚: 𝒢 × 𝒢 → P* (𝒢), which is defined as ⊚(𝓅,𝒹)=𝓅 ⊚𝒹, called “hyperoperation” [3], where P*(𝒢) is the set of all not empty subsets of 𝒢. An algebraic hyperstructure (𝒢,⊚) is referred to as a “hypergroupoid” [3]. This hypergroupaid is said to be “semihypergroup” if; 𝓅 ⊚(𝒹 ⊚𝓆) = (𝓅 ⊚𝒹)⊚𝓆, for 𝑎𝑙𝑙 𝓅,𝒹,𝓆 ∈ 𝒢, i,e: 𝑈𝑢∈𝒹⊚𝓆 𝓅 ⊚ 𝑢 = 𝑈𝑣∈𝓅⊚𝒹 𝑣 ⊚𝓆. Also, for any ℰ ≠ ∅ 𝑎𝑛𝑑 𝒞 ≠ ∅ subsets of 𝒢 and 𝓅 ∈ 𝒢, defined ℰ ⊚𝒞 = 𝑈ℯ∈ℰ.𝒸∈𝒞 ℯ ⊚𝒸, ℰ ⊚ 𝓅 = ℰ ⊚ {𝓅} and 𝓅 ⊚ 𝒞 = {𝓅} ⊚ 𝒞. [3], And (𝒢,⊚) is called “quasihypergroup”, if 𝒢 ⊚𝑥 = 𝑥 ⊚𝒢 = 𝒢,𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝒢, [1]. If the pair (𝒢,⊚) satisfied the conditions of the semihypergroup and the quasihypergroup, then called “hypergroup” [3]. A set doi.org/10.30526/36.2.3028 Article history: Received 21 September 2022, Accepted 21 November 2022, Published in April 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Some Results on the Divisible Hyperrings Mayssam Fadel Abood Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq. maisssam.fadel1203a@csw.uobaghdad.edu.iq Tamadher Arif Ibrahiem Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq. tamadherai_math@csw.uobaghdad.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:maisssam.fadel1203a@csw.uobaghdad.edu.iq mailto:tamadherai_math@csw.uobaghdad.edu.iq IHJPAS. 36(2)2023 384 ∅ ≠ 𝒬 that contained in (𝒢,⊚) is called “subhypergroup” if it was hypergroup it-self [3]. In 1956 Krasner introduced the concept of hyperring and hypermodule which are known nowadays as Krasner hyperring and Krasner hypermodule respectively. After that many authors introduced many types of hyperring like general hyperring and multiplicative hyperring. In this paper, the hyperring ℛ is a Krasner hyperring with unit element 1. 𝑅𝑜 is the set of all non-zero divisor elements of a hyperring ℛ, and a left ℛ-hypermodule will be denoted by ℳ. 2. Preliminaries. Definition 2.1 [4]. The hypergroup (𝒢,⊚) is called canonical if; 1. ⊚ is an associative hyperoperation, i,e 𝓅 ⊚(𝒹 ⊚𝓆) = (𝓅 ⊚𝒹)⊚ 𝓆 for every 𝓅,𝒹,𝓆 ∈ 𝒢; 2. There exist an element“0” ∈ 𝒢, such that 0⊚𝓅 = 𝓅 ⊚0 = {𝓅}.∀ 𝓅 ∈ 𝒢; 3. There exist a unique −𝓅 ∈ 𝒢,∀ 𝓅 ∈ 𝒢, such that 0 ∈ 𝓅 ⊚(−𝓅); 4. 𝓅 ∈ 𝒹 ⊚𝓆 implies 𝒹 ∈ 𝓅 ⊚(−𝓆). 5. For all 𝓅,𝒹 ∈ 𝒢, 𝓅 ⊚𝒹 = 𝒹 ⊚𝓅. Definition 2.2 [4] . The hyperstructure (ℛ ⊚,⊙ ) is said to be Krasner hyperring, if: 1. (ℛ,⊚) is a canonical hypergroup; 2. (ℛ,⊙) is a semigroup, have 𝛼⨀0 = 0⨀𝛼 = 0, for all 𝛼 ∈ ℛ 3. 𝛼⨀(𝛽 ⊚𝛾) = 𝛼⨀𝛽 ⊚𝛼⨀𝛾 𝑎𝑛𝑑 (𝛽 ⊚𝛾)⨀𝛼1 = 𝛽⨀𝛼 ⊚𝛾⨀𝛼. For all 𝛼,𝛽 𝑎𝑛𝑑 𝛾 ∈ ℛ. A Krasner hyperring is commutative if (ℛ,⨀) is a commutative semigroup. Definition 2.3 [5]. A subset 𝒜 of ℛ is said to be subhyperring if it satisfy the conditions of a hyperring. Definition 2.4 [5]. Let (ℛ,∔,⋅) be a commutative hyperring, 𝐼 be a hyperideal of ℛ, then the set ℛ/𝐼 = {𝑥 ∔𝐼:𝑥 ∈ ℛ} is a commutative hyperring under hyperaddition (𝑥 ∔ 𝐼 ) ∔ (𝑦 ∔ 𝐼) = (𝑥 ∔ 𝑦) ∔𝐼 and multiplication ( 𝑠 ∔ 𝐼)(𝑡 ∔ 𝐼) = 𝑠𝑡 ∔ 𝐼. And it is called a quotient hyperring. Definition 2.5 [1] . Let I be a nonempty subset of a Krasner hyperring ℛ, then I is called a “ right (resp. left) hyperideal” if for every 𝓅 and 𝒹 ∈ I, and 𝛼 ∈ ℛ: (1) 𝓅 −𝒹 ⊆ Ι; (2) 𝛼⨀𝓅 ∈ Ι (resp.𝓅⨀𝛼 ∈ I) And called hyperideal if it is right and left hyperideal. Definition 2.6 [4]. The hyperideal I of ℛ is maximal hyperideal if every hyperideal J in ℛ with I ⊊ J ⊆ ℛ then J = ℛ. Definition 2.7 [1]. A canonical hypergroup (ℳ,⨁) is said to be left hypermodule over a hyperring (ℛ,⊚,⨀) with the unit element “1”, if the map ⋅ : ℛ ×ℳ ⟶ ℳ which is defined as: ⋅ (𝑠,𝑚) ↦ 𝑠.𝑚 = 𝑠𝑚 ∈ ℳ, for 𝑠 ∈ ℛ,𝑚 ∈ ℳ Satisfies the following conditions, for 𝓀,𝒷 ∈ ℛ, and 𝑚ˋ ∈ ℳ: 1. (𝓀 ⊚𝒷)𝑚 = 𝓀𝑚⊕𝒷𝑚 IHJPAS. 36(2)2023 385 2. 𝓀(𝑚⨁𝑚ˋ)=𝓀𝓂 ⊕𝓀𝑚ˋ 3. (𝓀⨀𝒷)𝑚=𝓀(𝒷𝑚) 4. 0ℛ .𝑚 = 0ℳ, where 0ℛ is a zero of ℛ, 0ℳ the secular identity of ℳ. In the same way, one can define the right ℛ-hypermodule. The ℛ- hypermodule ℳ is said to be unitary if 1.𝓂 = 𝓂, where 1 is the unit element of ℛ and 𝓂∈ℳ. Definition 2.8 [6]. If ℳ is an ℛ-hypermodule, then ∅ ≠ 𝒩 ⊆ ℳ is called a “Subhypermodule” if and only if ℴ −𝒶 ⊆ 𝒩 and ℴ𝓇 ∈ 𝒩, for each ℴ, 𝒶 ∈ 𝒩, 𝓇 ∈ ℛ. Definition 2.9 [2] Let (ℛ,⊚,⨀) be a hyperring. The element 𝑟1 ∈ ℛ is named “right (resp. left)- zero divisor” if there is 0 ≠ 𝑟2 ∈ ℛ such that 𝑟1.𝑟2 (resp.𝑟2.𝑟1) = {0}. And called zero-divisor if it was right and left zero-divisor. Definition 2.10 [2] Let ℳ be an ℛ-hypermodule.The element 𝜇 ∈ ℳ is called a “divisible element” if for each non-zero divisor 𝜍 ∈ ℛ there is ∈ ℳ, such that. 𝜇 = 𝜍𝜂. If every element in an ℛ-hypermodule ℳ is a divisible, then ℳ is named a “divisible hypermodule” Definition 2.11 [4]. The Jacobson radical of a hyperring ℛ is the intersection of all maximal- hyperideals and it is denoted by J(ℛ). Remark 2.12 [7]. J(ℛ) is a hyperideal in ℛ Notations: • 𝑈𝑙𝑒𝑓𝑡 (ℛ) ={r 𝜖 ℛ | Ǝŕ 𝜖 ℛ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ŕ.𝑟 = 1} • 𝑈𝑟𝑖𝑔ℎ𝑡 (ℛ) = {𝑟 ϵ ℛ |Ǝ ŕ 𝜖 ℛ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑟.ŕ = 1} • 𝑈(ℛ) = { 𝑟𝜖ℛ |Ǝŕ 𝜖 ℛ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑟.ŕ = ŕ.𝑟 = 1} In the following proposition, Davvaz B and Salasi A in [4] proved the part (1 → 2). Here we add another condition and give the following proposition. Proposition 2.13. For an element 𝓅 ∈ ℛ the following statements are equivalent (1) 𝓅 ∈ J(ℛ) (2) for any 𝒹 ∈ ℛ, 1-𝓅𝒹 ⊆ 𝑈 (ℛ) (3) for any 𝒹,𝓆 ∈ ℛ, 1-𝓅𝒹𝓆 ⊆ 𝑈(ℛ). Proof. (1 ↔ 2 proved in (4, Prop. 2.14)). Now, will prove (1 ↔ 3). To prove that, for 𝓅 ∈ J(ℛ), and for all 𝒹,𝓆 ∈ ℛ, the set 1- 𝓅𝒹𝓆 ⊆ 𝑈(ℛ). Suppose that ∃𝒹𝑜,𝓆𝑜 ∈ ℛ such that 𝓈 ∉ 𝑈(ℛ), for some 𝓈 ∈ 1 - (𝒹𝑜𝓆𝑜) 𝓅. But 𝓈 ∉ 𝑈(ℛ) this lead to ℛ𝓈 ≠ ℛ then there is a maximal hyperideal 𝐼 such that ℛ𝓈 ⊆ 𝐼 (4, Prop.2.12), since 𝓈 ∈ 1 - (𝒹𝑜𝓆𝑜) 𝓅, one obtain that 1∈ 𝓈 +(𝒹𝑜𝓆𝑜) 𝓅 ⊆ 𝐼 +J(ℛ) ⊆ 𝐼 (by Definition 2.1) that is 1∈ 𝐼 and this contradiction. Conversely, for any 𝒹,𝓆 ∈ ℛ, let 1-𝓅𝒹𝓆 ⊆ 𝑈(ℛ). To prove that 𝓅 ∈ J(ℛ), assume 𝓅 ∉ J(ℛ), so ∃ 𝐼 is a maximal hyperideal of ℛ such that 𝓅 ∉ 𝐼. Thus <𝐼,𝓅> ≠ 𝐼, and so <𝐼,𝓅> = ℛ. Since 1 IHJPAS. 36(2)2023 386 ∈ ℛ, then there is 𝑎 ∈ 𝐼, and 𝒹,𝓆 ∈ ℛ such that 1∈ 𝑎 +𝓅(𝒹𝓆). Hence 𝑎 ∈1 - 𝓅(𝒹𝓆) ⊆ 𝑈(ℛ), this implies that 1 ∈ 𝐼, and this contradiction, therefore 𝓅 ∈ J(ℛ).∎ 3. Main results The concept of a divisible hyperring will be discussed in this section. Definition 3.1. The family of all divisible elements of an ℛ-hypermodule ℳ are defined as 𝑑(ℳ) ={ 𝑦 ∈ ℳ| for each 𝑟 ∈ 𝑅𝑂, there is 𝑥 ∈ ℳ, such that 𝑦 = 𝑟𝑥}. Remark 3.2. 𝑑(ℳ) is a divisible hypergroup of (ℳ,+). Proof. Let 𝑚1,𝑚2 ,𝑚3 ∈ 𝑑(ℳ). The associative of “+” is an obvious. Now, to prove 𝑚∘ 𝑑(ℳ) = 𝑑(ℳ)∘𝑚 = 𝑑(ℳ), let 𝑥 ∈ 𝑚 ∘ 𝑑(ℳ). It is mean 𝑥 ∈ 𝑚∘𝑦; 𝑦 ∈ d(ℳ) . For each 𝑟 ∈ 𝑅𝑜 there is 𝑧 ∈ ℳ such that 𝑦 = 𝑟𝑧; 𝑚 ∈ 𝑑(ℳ) 𝑚𝑒𝑎𝑛 𝑡ℎ𝑎𝑡 𝑚 = 𝑟𝑤,𝑤 ∈ ℳ. Follows 𝑥 ∈ (𝑟𝑤)∘(𝑟𝑧), lead to 𝑥 ∈ 𝑟(𝑤 ∘𝑧) ∈ 𝑑(ℳ). In the same way 𝑑(ℳ) ∘𝑚 = 𝑑(ℳ) can be proved. Proposition 3.3. If the hyperstructure (ℛ,⊚,⊙) is commutative hyperring then; 1. d(ℳ) is divisible ℛ-Subhypermodule of an ℛ-hypermodule ℳ 2. d(ℳ ∕d(ℳ)) = 0. Proof. 1. Let 𝑚,𝑛 ∈ d(ℳ), For any 𝑟 ∈ 𝑅𝑂, 𝑚 = 𝑟𝑚ˊ and 𝑛 = 𝑟𝑛ˊ. So 𝑚 −𝑛 = 𝑟𝑚ˊ−𝑟𝑛ˊ = 𝑟(𝑚ˊ−𝑛ˊ) ∈ d(ℳ). Now, if 𝑦 ∈ d(ℳ), 0 ≠ 𝑎 ∈ ℛ and for any r∈ 𝑅𝑂, ∃𝑥 ∈ ℳ such that, 𝑦 = 𝑟𝑥. Therefore, 𝑎𝑦 = 𝑎(𝑟𝑥) = (𝑎𝑟)𝑥 = (𝑟𝑎)𝑥 = 𝑟(𝑎𝑥). This implies that 𝑎𝑦 ∈ 𝑑(ℳ). 2. If y + d(ℳ) ∈ d(ℳ ∕d(ℳ)) for y∈d(ℳ). Then ∀ 𝑟 ∈ 𝑅𝑂,∃ x+d(ℳ) ∈ ℳ ∕d(ℳ) such that, y+d(ℳ) = r(x+d(ℳ)). Thus y-rx ∈ d(ℳ). This implies that ∃𝑥ˊ ∈ ℳ such that, (y-rx) = r𝑥ˊ. It follows that y=r(x+𝑥ˊ) and y ∈ d(ℳ) or equivalently y + d(ℳ) = d(ℳ), thus d(ℳ ∕d(ℳ)) = 0. Remark 3.4. The set d(ℛ) = {a ∈ ℛ| for all r ∈ 𝑅𝑂, there is b ∈ ℛ such that. a=rb} . Remark 3.5. The set d(ℛ) is a right hyperideal of ℛ. Indeed, for any y and x in d(ℛ), y =rs and x=ra, for all s and a belong to ℛ, y-x ⊆d(ℛ). Also for any r ∈ 𝑅𝑜 there is s ∈ ℛ such that, y=rs, thus yt = (rs)t = r(st), this implies yt ∈ d(ℛ), t∈ ℛ. Definition 3.6 [3] . A hyperring (ℛ,⊚,⊙) is a division hyperring if (ℛ\{0},⨀) is a hypergroup with unit element 1. Proposition 3.7. If ℛ is a division hyperring, then d(ℛ) = ℛ. Proof. It is clear that d(ℛ)⊆ ℛ. Now, for any a ∈ ℛ and any r ∈ 𝑅𝑂=ℛ∗, ∃ b = 𝑟−1a ∈ ℛ such that a = rb, therefore a ∈ d(ℛ). Proposition 3.8. Let ℛ be a hyperring that satisfies the property d(ℛ)∩𝑅𝑂 ≠ ∅, then: d(ℛ)= ∩ {𝑅ˆ| 𝑅ˆ is a left hyperideal of hyperring ℛ such that.𝑅ˆ ∩𝑅𝑂 ≠ ∅}. Proof. It’s clear that ∩{Rˆ| Rˆ is a left hyperideal in a hyperring ℛ, such that Rˆ∩𝑅𝑂≠∅} ⊆ d(ℛ). Now to prove the converse inclusion, let y ∈ d(ℛ) and 𝑅ˆ be the left hyperideal of a hyperring ℛ such that, 𝑅ˆ ∩𝑅𝑂 ≠ ∅ , let 𝑦𝑜 ∈ 𝑅 ˆ ∩𝑅𝑂, then for any y∈ d(ℛ), there is s ∈ ℛ such that, y =𝑦𝑜s ∈ 𝑅ˆ this gives d(ℛ) ⊆ 𝑅ˆ. Since 𝑅ˆ is an arbitrary hyperideal, so y ∈ ∩{Rˆ}. IHJPAS. 36(2)2023 387 Proposition 3.9. If ℛ is commutative hyperring and 𝑅𝑂 ∩𝐽(ℛ) ≠ ∅, then 1+d(ℛ) ⊆ 𝑈(ℛ). Proof. Let 𝑟𝑜 ∈ 𝑅 𝑂 ∩ 𝐽(ℛ) and for all 𝑟 ∈ d(ℛ), ∃𝑎 ∈ ℛ such that, 𝑟 = 𝑟𝑜𝑎. By (Prop.2.13-3), 1+𝑟 = 1+𝑟𝑜𝑎 = 1−1 ∙𝑟𝑜(−𝑎) ∈ 𝑈(ℛ) implies 1+𝑑(ℛ) ⊆ 𝑈(ℛ). Proposition 3.10. If ℛ is a hyperring satisfy 𝑅𝑂 = 𝑅*, then either d(ℛ)=0 or d(ℛ)=ℛ. Proof. If d(ℛ) ≠ 0, then for any 𝑦 in d(ℛ)\{0}, have 𝑦2 ≠ 0 and ∃ 𝑥 ∈ ℛ such that 𝑦 = 𝑦2𝑥, thus 𝑦(1−𝑦𝑥) = 0, this implies 𝑦𝑥 = 1 so, 𝑦 ∈ 𝑈𝑟𝑖𝑔ℎ𝑡(ℛ) (1) Also, 1−𝑦𝑥 = 0 leads to 𝑦−𝑦𝑥𝑦 = 0 coming after 𝑦(1−𝑥𝑦) = 0. But 𝑅𝑂 = 𝑅*, then 𝑥𝑦 = 1, thus 𝑦 ∈ 𝑈𝑙𝑒𝑓𝑡(ℛ) (2) Now by (1) and (2) get 𝑦 ∈ 𝑈(ℛ). Since 𝑦 ∈ 𝑈(ℛ)∩𝑑(ℛ) 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑑(ℛ) = ℛ. Definition 3.11. The hyperring ℛ is called “divisible hyperring” if any element belongs to ℛ is a divisible element. The following corollary comes straightaway from definition 3.11 and remark 3.4 Corollary 3.12. ℛ is divisible hyperring if and only if d(ℛ)=ℛ. Corollary 3.13. Every division hyperring is divisible hyperring Next, will refer to the class �̌� of 𝓡 having the properties: • 𝑑(ℛ) ≠ 0 • If {𝐼𝑗}𝑗∈𝐽 is a set of all maximal right hyperideals of a hyperring ℛ, then have 𝑑(ℛ)∩𝐼𝑗 = 0 , or 𝑑(𝑅)∩𝐼𝑗 ∩𝑅 𝑂 ≠ ∅, ∀𝑗 ∈ 𝐽. Theorem 3.14. A hyperring ℛ is a division if and only if a) ℛ is divisible hyperring b) ℛ ∈ �̆� c) 𝑅0 = 𝑈(ℛ) Proof. Let 𝐼𝑗 be a maximal right hyperideal of a hyperring ℛ, by (a), ℛ is divisible. This implies that d(ℛ)=ℛ. The condition (b) gives 𝐼𝑗=0 or 𝐼𝑗 ∩𝑅 𝑂 ≠ ∅ for maximal right hyperideal 𝐼𝑗 of ℛ. Finally, by (c), either 𝐼𝑗 = 0 or 𝐼𝑗 = ℛ, but 𝐼𝑗 ≠ ℛ, so 𝐼𝑗 = 0. For the other side, since ℛ is a division then by Corollary 3.13, ℛ is divisible hyperring. Now to prove (b), since d(ℛ)=ℛ therefore 𝑑(ℛ) ≠ 0, and if {𝐼𝑗}𝑗∈𝐽 is a set of all maximal right hyperideals of a hyperring ℛ, then 𝑑(𝑅)∩𝐼𝑗 ∩𝑅 𝑂 ≠ ∅. Finally, since ℛ is a division then every element is unite and so ℛ has no zero divisor hence 𝑅0 = 𝑈(ℛ), then (c) holds. ∎ Theorem 3.15. A divisible hyperring can be written as ℛ=xℛ for any x ∈ 𝑅𝑂. Proof. It is enough to show that ℛ ⊆ xℛ. Since ℛ is divisible hyperring, then for any 𝑎 ∈ ℛ there is 𝑏 ∈ ℛ such that 𝑎 = x𝑏 and so 𝑎 ∈ xℛ, thus ℛ ⊆ xℛ. IHJPAS. 36(2)2023 388 As the definition of the direct sum of two hyperideals in [8] and the direct sum of subhypermodules in [1], we will define the direct sum of two subhyperrings as follows; Definition 3.16. The direct sum of two subhyperrings 𝒮 𝑎𝑛𝑑 𝒯, is denoted by 𝒮⊕𝒯 such that for each element 𝓀 ∈ 𝒮⨁𝒯, there is unique elements 𝓈 ∈ 𝒮,𝓉 ∈ 𝒯, 𝓀 = 𝓈 +𝓉. Theorem 3.17. If 𝒮 and 𝒯 be subhyperrings of a hyperring ℛ, and 𝒮, 𝒯 are divisible, then 𝒮⨁𝒯 is divisible. Proof. Let 𝑘 ∈ 𝑆⨁𝒯 and 𝑟 ∈ 𝑅𝑂, ∃𝓈 ∈ 𝒮 and 𝓉 ∈ 𝒯 such that 𝑘 = 𝓈 +𝓉. Now, 𝒯 𝑎𝑛𝑑 𝒮 are divisible, so 𝒮 = 𝑥𝒮 and 𝒯 = 𝑥𝒯, for 𝑥 ∈ 𝑅𝑂. Hence 𝓈 = 𝑥𝓈1 and 𝓉 = 𝑥𝓉1 for some 𝓈1 ∈ 𝒮 𝑎𝑛𝑑 𝓉1 ∈ 𝒯. Thus 𝓀 = 𝓈 +𝓉 = 𝑥𝓈1 +𝑥𝓉1 = 𝑥(𝓈1 +𝓉1). If 𝑢 ∈ 𝓈1+𝓉1 ⊆ 𝒮 +𝒯. Then, 𝑘 = 𝑥𝑢. Definition 3.18 [2]. The function ℓ from the hyperring (ℛ ⊚,⊙) with 1ℛ into the hyperring (𝒮,∔ˈ,⋅ˈ), with 1𝒮, is a hyperring homomorphism if for each 𝓀,𝒷 ∈ ℛ, 1. ℓ(𝓀 ⊚ 𝒷) = ℓ(𝓀) ∔ˈ ℓ(𝒷) 2. ℓ(𝓀 ⊙𝒷)= ℓ(𝓀) ⋅ˈ ℓ(𝒷) 3. ℓ(1ℛ)= 1𝒮. Remark 3.19. A function ℓ is named as surjective ℛ-homomorphism if Im(ℓ) = 𝒮. Proposition 3.20. Let 𝒮 be a subhyperring of a divisible hyperring ℛ, then the quotient hyperring ℛ 𝒮 is a divisible hyperring Proof. Let 𝑎 +𝒮 ∈ 𝑑( ℛ 𝒮 ), so ∃𝑎ˊ ∈ ℛ such that 𝑎 = 𝑟𝑎ˊ for 𝑟 ∈ 𝑅𝑂. Thus 𝑎 +𝒮 = 𝑟𝑎ˊ +𝒮 = 𝑟(𝑎ˊ +𝒮), which implies ℛ 𝒮 is divisible hyperring. Corollary 3.21. Let 𝑓 be a surjective R-homomorphism where ℛ and 𝒮 are hyperrings. If ℛ is a divisible hyperring, then so is 𝒮. Examples 3.22. [2]. The hyperoperation “⊚” and the multiplication “ ⨀” are defined by the following in tables 1 and 2 on ℛ = {0, 1, 𝓅,𝒹}. Table 1: additive hyperoperayion ⊚ 0 1 𝓅 𝒹 0 {0} 1 𝓅 𝒹 1 1 ℛ {1.𝓅.𝒹} {1,𝓅,𝒹} 𝓅 𝓅 {1,𝓅,𝒹} ℛ {1,𝓅,𝒹} 𝒹 𝒹 {1,𝓅,𝒹} {1,𝓅,𝒹} ℛ IHJPAS. 36(2)2023 389 Table 2: multiplication Then ℛ is a hyperring and every nonzero element is a divisible, thus ℛ is divisible hyperring Conclusion. In this research, we discussed the concept of divisible hyperring. Some of the properties of divisible hyperring were studied. Clarify some concepts related to this concept. References. 1. Boriboon, S.; Pianskool, S., Baer hypermodules over Krasner hyperrings. 22ndAnnu. Meet Math. 2 Jun 2017, 1-9 2. Bordbar, H.; Cristea I., Divisible hypermodules. Analele Univ “Ovidius”, Constanta - Ser Mat. 2022, 30, 1, 57-74. 3. Harijani, KM.; Anvariyeh, SM., Non-Commutative Hypervaluation on Division Hyperrings. J Math Ext. 2019, 13, 2, 93–110. 4. Davvaz, B.; Salasi, A., A realization of hyperrings. Commun Algebra 2006, 34,12, 4389–400. 5. Abumghaiseeb, A. On 𝜹-Primary hyperideals and fuzzy hyperideals expansions, Republic of Turkey Yildiz Technical University Graduate School of Natural and Applied Sciences. 2018, PhD Thesis. 6. Bordbar, H.; Novák, M.; Cristea, I., A note on the support of a hypermodule. Journal of Algebra and Its Applications. 2020, 19,01, 2050019. 7. M,Anbarloei., J-prime hyperideals and their generalizations. arxiv:2110.07073. [math.AC]. 13 Oct 2021, https://doi.org/10.48550/arXiv.2110.07073 8. Yazarli, H.; Yılmaz, D.; Davvaz, B., The maximal hyperrings of quotients. Ital. J.Pure Appl. Math. 2020, 44, 938–951. ⨀ 0 1 𝓅 𝒹 0 0 0 0 0 1 0 1 𝓅 𝒹 𝓅 0 𝓅 𝒹 1 𝒹 0 𝒹 1 𝓅 https://www.worldscientific.com/doi/10.1142/S021949882050019X https://www.worldscientific.com/worldscinet/jaa https://www.worldscientific.com/worldscinet/jaa https://www.worldscientific.com/toc/jaa/19/01 https://arxiv.org/search/math?searchtype=author&query=Anbarloei%2C+M https://arxiv.org/abs/2110.07073 https://doi.org/10.48550/arXiv.2110.07073