EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6096 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Note on k-Ideals in Ternary Semirings Arbaz Jehan Khan1, Montakarn Petapirak1, Ronnason Chinram1,∗ 1Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand Abstract. In this paper, we give some examples of k-ideals of ternary semirings. We examine the results of k-ideals by distinguished classes in ternary semirings, which include k-maximal, k-prime and k-semiprime. 2020 Mathematics Subject Classifications: 16Y60,16Y99 Key Words and Phrases: Ternary semirings, k-ideals, k-maximal, k-prime, k-semiprime 1. Introduction Lehmer [1] introduced ternary algebra in 1932 and investigated certain algebraic sys- tems called triplexes, which are commutative ternary groups. Later, Banach also studied these algebraic structures and provided examples of a ternary semigroup that does not re- duce to a semigroup. Additionally, Lister [2] introduced the concept of a ternary ring. The concept of ternary semirings was first introduced by Dutta and Kar [3] in 2003. Moreover, Dutta and Kar investigated some basic concepts of prime ideals and semiprime ideals of ternary semirings in [4] and [5], respectively. The concept of ternary semirings arises from the study of algebraic structures that extend semirings. Ternary semirings consist of two operations, typically the addition and the ternary multiplication. By using a ternary mul- tiplication instead of a binary multiplication, every semiring can be turned to a ternary semiring. However, a ternary semiring does not necessarily reduce to a semiring. Although ternary semirings generalize the notion of a semiring, they are not just a generalization because some certain notions, such as lateral ideals, lack an analog in a semiring. Many concepts from semiring theory were extended to the study of ternary semirings. Ideal theory is the main area of research in the study of many algebraic structures. In 2005, Kar [6] introduced the notions of quasi-ideals and bi-ideals in ternary semirings and characterized regular ternary semirings in terms of quasi-ideals and bi-ideals. In 2010, Malee and Chinram studied fuzzifications of some ideals in ternary semirings [7] and [8]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6096 Email addresses: arbazjehankhan@gmail.com (A. J. Khan), montakarn.p@psu.ac.th (M. Petapirak) , ronnason.c@psu.ac.th (R. Chinram) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 2 of 10 The next year, Dubey [9] investigated quasi k-ideals and bi k-ideals in ternary semirings. In the same year, Chaudhari and Ingale [10] introduced a partitioning ideal of a ternary semiring and Sunitha et al. [11] provided the characterization of full k-ideals in ternary semirings. Partitioning ideals are useful to develop the quotient structures of ternary semirings. Singular ideals were introduced in [12] in 2012. In 2016, the concept of the subtractive extension of an ideal of a ternary semiring was introduced in [13]. In 2024, Luangchaisri and Changphas [14] studied right weakly regular ternary semirings and fully prime right ternary semirings. In the same year, Goswami and Dube [15] investigated some aspects of k-ideals of semirings. This paper inspired by their work. Our goal is to explore various aspects of k-ideals in ternary semirings. 2. Preliminaries In this section, we will recall some basic definitions of ternary semirings. Definition 1 ([3]). A nonempty set R together with a binary operation, called the ad- dition, and the ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring if R is an additive commutative semigroup satisfying the following conditions: (i) (abc)de = a(bcd)e = ab(cde), (ii) (a+ b)cd = acd+ bcd, (iii) a(b+ c)d = abd+ acd, (iv) ab(c+ d) = abc+ abd, for all a, b, c, d, e ∈ R. Example 1. (1) Every semiring can be considered a ternary semiring under the ordinary addition and ternary multiplication of semirings. (2) Z− is a ternary semiring under the usual addition and ternary multiplication of integers, but it is not a semiring under the usual addition and binary multiplication of integers. Definition 2 ([3]). Let R be a ternary semiring. If 0 ∈ R such that 0 + x = x and 0xy = x0y = xy0 = 0 for all x, y ∈ R, then 0 is called a zero element. In this case, R is called a ternary semiring with zero. Definition 3. A ternary semiring R is said to be commutative if abc = bca = cab = bac = cba = acb for all a, b, c ∈ R. Definition 4 ([3]). An additive semigroup S of a ternary semiring R is called a ternary subsemiring of R if s1s2s3 ∈ S for all s1, s2, s3 ∈ S. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 3 of 10 Definition 5. An additive subsemigroup I of a ternary semiring R is called (1) a left ideal of R if r1r2a ∈ I for all r1, r2 ∈ R and a ∈ I, (2) a right ideal of R if ar1r2 ∈ I for all r1, r2 ∈ R and a ∈ I, (3) a lateral ideal of R if r1ar2 ∈ I for all r1, r2 ∈ R and a ∈ I, (4) an ideal of R if I is a left ideal, a right ideal, and a lateral ideal of R. An ideal I of R is called a proper ideal if I ̸= R. Proposition 1 ([3]). Let R be a ternary semiring and a ∈ R. Then the following state- ments hold. (1) The principal left ideal generated by a is given by ⟨a⟩l = N0a+RRa. (2) The principal right ideal generated by a is given by ⟨a⟩r = N0a+ aRR. (3) The principal lateral ideal generated by a is given by ⟨a⟩m = N0a+RaR+RRaRR. (4) The principal ideal generated by a is given by ⟨a⟩ = N0a + RRa + aRR + RaR + RRaRR. If R is commutative, we note that ⟨a⟩ = N0a+RRa. Definition 6. An ideal I of a ternary semiring R is called a k-ideal if, for all x, y ∈ R, x ∈ I and x+ y ∈ I imply y ∈ I. Definition 7. A proper ideal of a ternary semiring R is called maximal if it is not properly contained in any other proper ideal of R. Definition 8. ([4]) A proper ideal P of a ternary semiring R is called prime if IJK ⊆ P implies I ⊆ P , J ⊆ P , or K ⊆ P for all ideals I, J,K of R. Definition 9. ([5]) A proper ideal P of a ternary semiring R is called semiprime if I3 ⊆ P implies I ⊆ P for every ideal I of R. 3. Main Results Let J (R) denote the set of all ideals of a ternary semiring R and JK(R) denote the set of all k-ideals of R. Suppose R is a commutative ternary semiring with zero 0. If A is a nonempty subset of R, then the annihilator of A is defined by AnnR(A) = {r ∈ R | rxy = 0 for all x, y ∈ A}. Proposition 2. Let R be a commutative ternary semiring with zero 0. For any two nonempty subsets A and B of R, if A ⊆ B, then AnnR(B) ⊆ AnnR(A). A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 4 of 10 Proof. Let r ∈ AnnR(B). So rxy = 0 for all x, y ∈ B. This implies rxy = 0 for all x, y ∈ A. Hence r ∈ AnnR(A). Example 2. Let R = {0, 2, 4, 6, 8, 10, 12, 14} ⊆ Z16. We have that R is a commutative ternary semiring under the usual addition and ternary multiplication of integers modulo 16. The element 0 is a zero of R. (1) AnnR(R) = {0, 4, 8, 12}. (2) If A = {4}, then AnnR(A) = R. (3) If A = {2}, then AnnR(A) = {0, 4, 8, 12}. Proposition 3. Let R be a commutative ternary semiring with zero 0 and a ∈ R. Then AnnR({a}) = AnnR(⟨a⟩). Proof. By Proposition 2, we have AnnR(⟨a⟩) ⊆ AnnR({a}). Let r ∈ AnnR({a}). Then raa = 0. Next, let x, y ∈ ⟨a⟩. So x = ka + ∑ sis ′ ia and y = k′a + ∑ rjr ′ ja for some k, k′ ∈ N0 and si, s ′ i, rj , r ′ j ∈ R. Since R is commutative, it is easy to see that rxy = 0. Hence AnnR({a}) = AnnR(⟨a⟩). Proposition 4. Let R be a commutative ternary semiring with zero 0 and A be any nonempty subset of R. Then AnnR(A) is a k-ideal of R. Proof. Clearly, 0 ∈ AnnR(A), which implies that AnnR(A) ̸= ∅. Let r, s ∈ AnnR(A) and x, y ∈ R. Then rab = 0 and sab = 0 for all a, b ∈ A, so (r+ s)ab = 0 and (rxy)ab = 0 for all a, b ∈ A. Then r + s ∈ AnnR(A) and rxy ∈ AnnR(A). This implies that AnnR(A) is an ideal of R. Let r, x ∈ R be such that r + x ∈ AnnR(A) and r ∈ AnnR(A). So (r+x)ab = 0 and rab = 0 for all a, b ∈ A. Then xab = 0+xab = rab+xab = (r+x)ab = 0. This implies that x ∈ AnnR(A), and hence AnnR(A) is a k-ideal of R. Let R be any ternary semiring. The k-closure operation on J (R) is defined for an ideal I of R by Ck(I) = {r ∈ R | r + x ∈ I for some x ∈ I}. Example 3. We consider a ternary semiring Z− 0 under the usual addition and ternary multiplication of integers. (1) Let I = 2Z− 0 ∖ {−2} = {0,−4,−6,−8,−10, . . .}. It is easy to show that I is an ideal of Z− 0 . We have Ck(I) = {0,−2,−4,−6,−8, . . .} = 2Z− 0 . (2) Let I = Z− 0 ∖ {−1} = {0,−2,−3,−4,−5, . . .}. It is easy to prove that I is an ideal of Z− 0 . We have Ck(I) = Z− 0 . Proposition 5. Let R be a ternary semiring and I be an ideal of R. Then Ck(I) is the smallest k-ideal containing I. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 5 of 10 Proof. Let x ∈ I. So x + x ∈ I. Then x ∈ Ck(I). So I ⊆ Ck(I). Let r, s ∈ Ck(I). Then there exist x, y ∈ I such that r + x ∈ I and s + y ∈ I. Then r + s + x + y ∈ I. This implies that r + s ∈ Ck(I). Next, let r ∈ Ck(I) and a, b ∈ R. Thus there exists x ∈ I such that r + x ∈ I. Then rab + xab = (r + x)ab ∈ I and xab ∈ I. This shows that rab ∈ Ck(I). Similarly, arb, abr ∈ Ck(I). Then Ck(I) is an ideal of R. Next, assume that r + x, x ∈ Ck(I). There exist a, b ∈ I such that r + x + a, x + b ∈ I. Then x + a + b ∈ I and r + x+ a+ b ∈ I. Hence r ∈ Ck(I). We can conclude that Ck(I) is a k-ideal of R. For the next step, let J be any k-ideal containing I. We need to show that Ck(I) ⊆ J . Let r ∈ Ck(I). Then there exists x ∈ I such that r + x ∈ I. Since I ⊆ J , r + x ∈ J and x ∈ J . Since J is a k-ideal such that r + x ∈ J and x ∈ J , it follows that r ∈ J . Thus Ck(I) ⊆ J . Therefore Ck(I) is the smallest k-ideal containing I. Proposition 6. Let R be a ternary semiring. The following statements hold. (1) Ck(R) = R. (2) If R has a zero, then Ck({0}) = {0}. Proof. (1) By Proposition 5, we have R ⊆ Ck(R). Then Ck(R) = R. (2) Let x ∈ Ck({0}). So x+ 0 ∈ {0}. This implies that x = 0 and Ck({0}) = {0}. Proposition 7. Let R be a ternary semiring. If I and J are any ideals of R such that I ⊆ J , then Ck(I) ⊆ Ck(J). Proof. Let r ∈ Ck(I). Then by definition of Ck(I), we have r + x ∈ I for some x ∈ I. Since I ⊆ J , r + x ∈ J and x ∈ J . Thus r ∈ Ck(J). Therefore Ck(I) ⊆ Ck(J). Theorem 1. Let R be a ternary semiring and I be an ideal of R. Then I is a k-ideal of R if and only if I = Ck(I). Proof. Let I be a k-ideal of R. By Proposition 5, we have I ⊆ Ck(I). Let r ∈ Ck(I). Then there exists x ∈ I such that r+ x ∈ I. Since I is a k-ideal of R and r+ x, x ∈ I, we obtain r ∈ I. Thus Ck(I) ⊆ I. We conclude that Ck(I) = I. Conversely, we assume that Ck(I) = I. By Proposition 5, I is a k-ideal of R. Proposition 8. Let I be an ideal of a ternary semiring R. Then Ck(Ck(I)) = Ck(I). Proof. By Proposition 5, we know that Ck(I) is a k-ideal of R. By Theorem 1, we have that Ck(Ck(I)) = Ck(I). Proposition 9. Let I, J and K be ideals of a ternary semiring R. Then Ck(I)Ck(J)Ck(K) ⊆ Ck(IJK). A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 6 of 10 Proof. Let x ∈ Ck(I), y ∈ Ck(J) and z ∈ Ck(K). Then there exist a ∈ I, b ∈ J and c ∈ K such that x + a ∈ I, y + b ∈ J and z + c ∈ K. So abc ∈ IJK and each of the following is also a member of IJK: xbc+ abc = (x+ a)bc ∈ IJK, (1) ayc+ abc = a(y + b)c ∈ IJK, (2) abz + abc = ab(z + c) ∈ IJK, (3) xyc+ ayc+ xbc+ abc = (x+ a)(y + b)c ∈ IJK, (4) ayz + abz + ayc+ abc = a(y + b)(z + c) ∈ IJK, (5) xbz + xbc+ abz + abc = (x+ a)b(z + c) ∈ IJK, (6) and xyz + xyc+ xbz + xbc+ ayz + ayc+ abz + abc = (x+ a)(y + b)(z + c) ∈ IJK. (7) Since abc ∈ IJK and xbc+abc ∈ IJK, we obtain xbc ∈ Ck(IJK). By the same argument, from (2) and (3), we also obtain ayc ∈ Ck(IJK) and abz ∈ Ck(IJK), respectively. Again, since abc ∈ IJK and (4) holds, we get xyc+ ayc+ xbc ∈ Ck(IJK). By Proposition 5, we know that Ck(IJK) is a k-ideal of R, which implies that xyc ∈ Ck(IJK) because xbc, ayc ∈ Ck(IJK), Applying the same process to (5) and (6), we conclude that ayz, xbz ∈ Ck(IJK), respectively. Once again, we see that (7) implies xyz + xyc+ xbz + xbc+ ayz + ayc+ abz ∈ Ck(IJK). Since xbc, ayc, abz, xyc, ayz and xbz are all members of Ck(IJK), we eventually obtain xyz ∈ Ck(IJK). Therefore, Ck(I)Ck(J)Ck(K) ⊆ Ck(IJK). Suppose that I, J and K are ideals of a ternary semiring R. Then the ideal quotient of I over J and K is defined by (I : J,K) = {a ∈ R | aJK ⊆ I}. Example 4. Consider R = {0, 2, 4, 6, 8, 10, 12, 14} ⊆ Z16 which is a ternary semiring under the usual addition and ternary multiplication of integers modulo 16. Let I = {0, 8}. We have that (I : J,K) = R for all ideals J and K of R. Proposition 10. Let I, J and K be ideals of a ternary semiring R. Then I ⊆ (I : J,K). Proof. Let a ∈ I. Since I is an ideal of R and J,K ⊆ R, we have that aJK ⊆ I. This implies that I ⊆ (I : J,K). Proposition 11. Let R be a commutative ternary semiring. If I is a k-ideal and J,K are ideals of R, then (I : J,K) is a k-ideal of R. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 7 of 10 Proof. Let a, b ∈ (I : J,K) and r, s ∈ R. Then aJK ⊆ I and bJK ⊆ I. Thus (a + b)JK ⊆ I and (ars)JK ⊆ a(rsJ)K ⊆ aJK ⊆ I. So a + b ∈ (I : J,K) and ars ∈ (I : J,K). Hence (I : J,K) is an ideal of R. Next, let a, b ∈ R be such that a ∈ (I : J,K) and a+ b ∈ (I : J,K). This implies that aJK + bJK = (a + b)JK ⊆ I and aJK ⊆ I. Let x ∈ bJK and y ∈ aJK. So x + y ∈ I and y ∈ I. Since I is a k-ideal of R, we have x ∈ I. This implies that bJK ⊆ I, so b ∈ (I : J,K). Hence (I : J,K) is a k-ideal of R. A proper ideal of R is called k-maximal if it is not properly contained in any other proper k-ideal of R. Example 5. We consider a ternary semiring R = Z− 0 under the usual addition and ternary multiplication of integers. (1) Let I = 2Z− 0 = {0,−2,−4,−6,−8,−10, . . .}. Then I is a k-ideal of R. We have that I is k-maximal but not maximal. (2) Let I = Z− 0 ∖ {−1} = {0,−2,−3,−4,−5,−6, . . .}. Then I is an ideal of R but not a k-ideal. We have that I is both maximal and k-maximal. Theorem 2. Let R be a ternary semiring and let I be a proper ideal of R. (1) If I is k-maximal, then I is a k-ideal or Ck(I) = R. (2) If I is a k-ideal and a maximal ideal, then I is k-maximal. Proof. (1) Suppose I is k-maximal. By Proposition 5, I ⊆ Ck(I) ⊆ R and Ck(I) is a k-ideal of R. This implies that Ck(I) = I or Ck(I) = R. Hence I is a k-ideal or Ck(I) = R. (2) Let I be a k-ideal and a maximal ideal. Suppose to the contrary, that I is not k-maximal. Then there exists a k-ideal J of R such that I ⊊ J ⊊ R. However, since I is a maximal ideal and J is an ideal of R, no such an ideal J exists unless J = R, which contradicts the fact that J ⊊ R. Therefore I must be k-maximal. A proper ideal P of a ternary semiring R is called k-prime if IJK ⊆ P implies I ⊆ P , J ⊆ P , or K ⊆ P for all k-ideals I, J,K of R. Every prime ideal of R is obviously k-prime. We describe in the following theorem, where being prime and k-prime of an ideal always imply each other. Theorem 3. Let R be a ternary semiring and P be a k-ideal of R. Then P is k-prime if and only if P is prime. Proof. If P is prime, then it is clear that P is k-prime. Assume that P is a k-prime k-ideal of R. Let I, J and K be ideals of R such that IJK ⊆ P . By Proposition 9 and Theorem 1, we have Ck(I)Ck(J)Ck(K) ⊆ Ck(IJK) ⊆ Ck(P ) = P. This implies that Ck(I)Ck(J)Ck(K) ⊆ P. Since P is k-prime and Ck(I), Ck(J) and Ck(K) are the smallest k-ideals containing I, J andK, respectively, we obtain that I ⊆ Ck(I) ⊆ P , J ⊆ Ck(J) ⊆ P , or K ⊆ Ck(K) ⊆ P . Therefore, P is prime. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 8 of 10 Proposition 12. Let R be a commutative ternary semiring and P be a proper ideal of R. If P is prime, then P = (P : R,R). Proof. For any prime ideal P of R, by Proposition 10, we have that P ⊆ (P : R,R). Assume that P ̸= (P : R,R). Then there exists x ∈ (P : R,R) but x /∈ P . So xRR ⊆ P . This implies that ⟨x⟩RR ⊆ P . However, since x /∈ P , we have that ⟨x⟩ ⊈ P which is a contradiction because P is prime. This forces P = (P : R,R). Proposition 13. Let R be a commutative ternary semiring and P be a proper ideal of R. If P is k-prime, then P = (P : R,R). Proof. It is similar to Proposition 12. A subset A of R is called a multiplicatively closed set if abc ∈ A for all a, b, c ∈ A. Theorem 4. Let P be a proper k-ideal of a ternary semiring R. Then P is k-prime if and only if R∖ P is a multiplicatively closed set. Proof. Assume that P is k-prime. First of all, we will show thatR∖P is multiplicatively closed, we thus let a, b, c ∈ R∖P . To show that abc ∈ R∖P , we suppose for contradiction, that abc ∈ P . Since P is k-prime, this implies a ∈ P , b ∈ P , or c ∈ P . This contradicts the fact that a, b, c ∈ R ∖ P . Thus abc ∈ R ∖ P and R ∖ P is multiplicatively closed. Conversely, assume that R ∖ P is multiplicatively closed. We need to show that P is k-prime. Let I, J,K be k-ideals of R such that IJK ⊆ P . We need to show that I ⊆ P , J ⊆ P , or K ⊆ P . Suppose to the contrary that there exist a ∈ I, b ∈ J and c ∈ K such that a, b and c are not members in P . Then we obtain that abc ∈ IJK and a, b, c ∈ R∖P . Since R ∖ P is multiplicatively closed, we then obtain abc ∈ R ∖ P . This contradicts abc ∈ IJK ⊆ P . Hence a ∈ P , b ∈ P , or c ∈ P . So P is k-prime. A proper ideal Q of R is called k-semiprime if I3 ⊆ Q implies I ⊆ Q for every k-ideal I of R. We have some remarks as the following. (1) Every prime ideal is semiprime. (2) Every k-prime ideal is k-semiprime. (3) Every semiprime ideal is k-semiprime. Theorem 5. Let R be a ternary semiring and Q be a k-ideal of R. Then Q is k-semiprime if and only if Q is semiprime. Proof. If Q is semiprime, then Q is clearly k-semiprime. Let Q be a k-semiprime k-ideal of R and assume that I3 ⊆ Q for an ideal I of R. Then Ck(I)Ck(I)Ck(I) ⊆ Ck(I3) ⊆ Ck(Q) = Q. Then Ck(I)Ck(I)Ck(I) ⊆ Q. Since Q is k-semiprime and Ck(I) is the smallest k-ideal containing I, we get I ⊆ Q. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 9 of 10 4. Conclusion In this paper, we focus on various properties of k-ideals of a ternary semiring R. We show that AnnR(A) is a k-ideal of R for every nonempty subset A of R. For an ideal I of R, we prove that Ck(I) is the smallest k-ideal containing I. In particular, for a k-ideal I and ideals J,K of a commutative ternary semiring R, we demonstrate that (I : J,K) is a k-ideal of R. Finally, we explore the relationship between k-ideals and k-maximal, k-prime and k-semiprime ideals. In future work, we can study other types of ideals and their properties in ternary semirings. Acknowledgements We sincerely appreciate all valuable comments and suggestions of reviewers, which helped us to improve the quality of the manuscript. This work was supported in part by the PSU-TUYF Charitable Trust Fund, Prince of Songkla University, Contract no.1-2567-01. References [1] D. H. Lehmer. A ternary analogue of abelian groups. Amer. J. Math., 54(2):329–338, 1932. [2] W. G. Lister. Ternary rings. Trans. Amer. Math. Soc., 154:37–55, 1971. [3] T. K. Dutta and S. Kar. On regular ternary semirings. Advances in Algebra, Proceed- ings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific, pages 343–355, 2003. [4] T. K. Dutta and S. Kar. On prime ideals and prime radical of ternary semirings. Bull. Cal. Math. Soc., 97(5):445–454, 2005. [5] T. K. Dutta and S. Kar. On semiprime ideals and irreducible ideals of ternary semir- ings. Bull. Cal. Math. Soc., 97(5):467–476, 2005. [6] S. Kar. On quasi-ideals and bi-ideals in ternary semirings. Int. J. Math. Math. Sci., 2005:3015–3023, 2005. [7] R. Chinram and S. Malee. L-fuzzy ternary subsemirings and l-fuzzy ideals in ternary semirings. IAENG Int. J. Appl. Math., 40(3):40 3 03, 2010. [8] S. Malee and R. Chinram. k-fuzzy ideals of ternary semirings. Int. J. Comp. Math. Sci., 4(4):206–210, 2010. [9] M. K. Dubey. A note on quasi k-ideals and bi k-ideals in ternary semirings. Italian J. Pure Appl. Math., 28:143–150, 2011. [10] J. N. Chaudhari and K. J .Ingale. On partitioning and subtractive ideals of ternary semirings. Kyungpook. Math. J., 51(1):69–76, 2011. [11] T. Sunitha; U. Nagi Reddy and G. Shobhalatha. A note on full k-ideals in ternary semirings. Indian J. Sci. Techno., 14(21):1786–1790, 2021. A. J. Khan, M. Petapirak, R. Chinram / Eur. J. Pure Appl. Math, 18 (2) (2025), 6096 10 of 10 [12] T. K. Dutta; K. P. Shum and S. Mandal. Singular ideals of ternary semirings. Eur. J. Pure Appl. Math., 5(2):116–128, 2012. [13] J. N. Chaudhari and K. J. Ingale. Subtractive extension of ideals in ternary semirings. Thai J. Math., 14(3):615–625, 2016. [14] P. Luangchaisri and T. Changphas. Prime one-sided ideals in ternary semirings. Int. J. Math. Comp. Sci., 19(2):403–409, 2024. [15] A. Goswami and T. Dube. Some aspects of k-ideals of semirings. Rend. Circ. Mat. Palermo (2), 73:3105–3117, 2024.