Malaya Journal of Matematik, Vol. 9, No. 1, 542-546, 2021 https://doi.org/10.26637/MJM0901/0091 Prime bi-interior Γ-ideals of TG-semring A. Nagamalleswara Rao1 , P. L. N. Varma2 , G. Srinivasa Rao3 *, D. Madhusudhana Rao4 and Ch. Ramprasad5 Abstract The notation of prime bi-interior ideal, semi prime bi-interior ideal, irreducible bi-interior ideal and strongly prime bi-interior ideal of TG-Semi ring (ternary gamma semi-ring) are introduced. We study properties of these ideals and relations between them and also characterize regular TG-Semi-ring and TG-Semi-ring using prime bi-interior ideals, irreducible and strongly irreducible bi-interior ideals in this article. Keywords TGS, bi-interior ideal, prime ideals, prime bi-interior ideal, strong prime bi-interior ideal, semi prime bi-interior ideal and irreducible and strongly irreducible bi-interior ideals. AMS Subject Classification 16W25, 16N60, 16U80. 1 Department of Mathematics, Acharya Nagarjuna University, Namburu, Guntur-522510, Andhra Pradesh, India. of Sciences & Humanities, VFSTR Deemed to be University, Vadlamudi, Guntur-522213, Andhra Pradesh, India. 4 Department of Mathematics, VSR & NVR Degree College, Tenali-522201, Andhra Pradesh, India. 5 Department of Sciences & Humanities, VVIT, Namburu, Guntur-522508, Andhra Pradesh, India. *Corresponding author: 3 gsrinulakshmi77@gmail.com c 2021 MJM. Article History: Received 11 December 2020; Accepted 12 February 2021 2,3 Department Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 Prime Bi-interior ideals in T gamma Semi rings. . . . .2 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Introduction We introduced the notation of prime bi-interior ideals of TG Semi rings, in this paper. G. Srinivasa Rao et.al [29-36] studied ternary semi rings. 2. Preliminaries Let (R, +)&(Γ, +), commutative semi-groups. Then we call R a T G -Semi ring (T GS), if there is a mapping R × Γ × R × Γ × R → R (images of (p, a, q, b, r) will be denoted by pagbr )∀p, q, r ∈ R, a, b ∈ Γ ∋ it satisfies the following axioms ∀p, q, r, s,t ∈ R and a, b, c, d ∈ Γ 1. pa(q + r)bs = pagbs + parbs 2. (p + q)arbs = parbs + qarbs 3. paqb(r + s) = paqbr +paqbs 4. pa(qbrcs)dt= paqb(rcsdt) =(paqbr)csdt. ATGSR is said to be commutative TGS if paqbr = parbq = qarbp = qapbr = rapbq = raqbp, ∀p, q, r ∈ R and a, b ∈ Γ. Suppose R, a TGS. If for each p in R∃a, b ∈ Γ papbe = paebp = eapbp = e, then an element e ∈ R is called aunityelement or neutral element. Definition 2.1. Suppose R, a ternary Γ -semi-ring. 0/ 6= S is said to be a ternary sub − Γ - semi-ring of R, if S is an subsemi-group with respect to + of R and aαbβ c ∈ S, ∀a, b, c ∈ S and α β ∈ Γ. Definition 2.2. The set 0/ 6= I ⊆ R, where R is a temary Γ semi-ring is said to be left (lateral, right)ternary Γ -ideal of R, if 1. a, b ∈ I ⇒ a + b ∈ I 2. a, b ∈ R, i ∈ I, α, β ∈ Γ ⇒ aαbβ i ∈ I(aαiβ b ∈ I, ia aβ b ∈ I). An ideal I is said to be ternary Γ -ideal, if it is left, lateral and right Γ -ideal of R Prime bi-interior Γ-ideals of TG-semring — 543/5 Example 2.3. The set Z = {0, ±1, ±2, ±3, . . . .} and Γ, the set of even natural numbers. Then with respect to usual addition and ternary multiplication, Z is a ternary gamma semi ring. Example 2.4. The set R = {0, ±i, ±2i, ±3i, . . . .} and Γ = N. Then R is a ternary gamma semi ring. with respect to usual addition and ternary multiplication. 3. Prime Bi-interior ideals in T gamma Semi rings We study ”the notion of prime, strongly prime, semi prime, irreducible and strongly irreducible bi-interior ideals” of TGsemi-rings. And we study the properties of prime ideals and relations between them. Definition 3.1. A BII B of a TGS R is said to a prime biinterior ideal (PBII) of R if B1 ΓB2 ΓB3 ⊆ B ⇒ B1 ⊆ B or B2 ⊆ B or B3 ⊆ B. Example 2.5. Let Q = R, the set of all rational numbers and Γ the set of all natural numbers. Define a mapping from R × Γ × R × Γ × R → R by usual addition and ternary multiplication defined by (p, a, q, b, r) = paqbr, ∀p, q, r ∈ R, a, b ∈ Γ then R is a ternary gamma semi-ring. Definition 3.2. A BII P of TGS R is called a semi prime biinterior ideal (SPBII) of R if for any bi- interior ideal P1 of R, P1 ΓP1 ΓP1 ⊆ P ⇒ P1 ⊆ P. Definition 2.6. Let φ 6= S ⊆ R, where R be a TGS. The set S said to be a T G -sub-semi-ring of R if (S, +) is a T sub-semigroup (T SSG) of (R, +) and SΓSΓS ⊆ S. Definition 3.3. A BII P of R is called a strongly prime biinterior ideal (ST PBII) if (P1 ΓP2 ΓP3 ) ∩ (P1 ΓP2 ΓP3 )∩(P1 ΓP2 ΓP3 ) ⊆ P ⇒ P1 ⊆ P or P2 ⊆ P or P3 ⊆ P, for any BI-ideals P1 , P2 and P3 of R. Definition 2.7. Let R be a TGS and φ 6= S ⊆ R. The set S said to be a quasi-ideal (QI) of R if S is a TGsub-semiring (TGSSR) of R and SΓRΓR ∩(RΓSΓR + RΓRΓSΓRΓR) ∩ RΓRΓS ⊆ S. Definition 3.4. A bi-interior ideal (BII) P of R is said to be an irreducible bi-interior ideal (IBII) P if BIIsP1 , P2 , P3 and P1 ∩ P2 ∩ P3 = P ⇒ P1 = P or P2 = P or P3 = P. Definition 2.8. Let φ 6= S ⊆ R, where R be a TGS. The set S said to be a bi-ideal (BI) of R if S is a TGSSR of R and SΓRΓSΓR Γ ⊆ S. Definition 3.5. A BII P of R, known as a strongly irreducible bi-interior ideal (SIBI) if for any BIs P1 , P2 and P3 of R, P1 ∩ P2 ∩ P3 ⊆ P ⇒ P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Clearly every SIBI is an IBI. Definition 2.9. Let φ 6= S ⊆ R, where R be a TGS. The set S said to be an interior ideal (II) of R if S is a TGSSR of R and RΓRΓSΓRΓR ⊆ S. Note 3.6. (ii) Every PBII P of a TGS R is a SPBII of R. Definition 2.10. Let R be a TGS and φ 6= S ⊆ R. The set S said to be a rt. (medial, lt.) ideal of R if S is a TGSSR of R and SΓRΓR ⊆ S(RΓSΓR ⊆ S, RΓRΓS ⊆ S). Theorem 3.7. A BII P of a T GSR is a prime bi-interior ideal ⇔ KΓMΓL ⊆ P ⇒ K ⊆ P or M ⊆ P or L ⊆ P where M is a lateral ideal, K is a right ideal and L is a left ideal of R. Definition 2.11. Let φ 6= S ⊆ R, where R be a TGS. The set S said to be an ideal if S is a TGSSR of R and SΓRΓR ⊆ S, RΓSΓR ⊆ S, RΓRΓS ⊆ S. Definition 2.12. Let φ 6= S ⊆ R, where R be a TGS. The set P said to be a bi-interior ideal (BII) of R if P is a TG subsemi ring (GSSR) of R ΓRΓPΓRΓR ∩ PΓRΓPΓRΓP ⊆ P. Definition 2.13. Let φ 6= S ⊆ R, where R be a TGS. The set P said to be a lt.(medial, rt. ) weak-interior ideal of R if P is a GSSR of R and RΓPΓP ⊆ P(PΓRΓP ⊆ P, PΓPΓR ⊆ P). Example 2.14. Consider the TGsemi-ring R = M2×2 ( W ), and Γ = M2×2 ( W ) where W = {0, 1, 2, 3, . . . . . .} Then R is a TG-semi-ring with PαQβ S is the ordinary ternary multiplication of matrices, ∀ P, Q, S ∈ R and α, β ∈ Γ. Here    p a U= : a, b, p ∈ W 0 b is a bi-ideal of R. Also  p V= 0 is a bi-ideal of R. 0 a  : a, p ∈ W  (i) Every STPBII of a TGS R is a PBII of R. Proof. Suppose that a prime bi-interior ideal P of the TGS R and KΓMΓL ⊆ P. Since K, M and L are bi- interior ideals of R, we have K ⊆ P or M ⊆ P or L ⊆ P. Again let KΓMΓL ⊆ P where K is a right ideal, M is a lateral ideal and L is a left ideal of R. ⇒ K ⊆ P or M ⊆ P or L ⊆ P. Suppose DΓEΓF ⊆ P, D, E and F are bi-interior ideals and (d)r, (e)m ( f )l are right, medial or lateral and left ideals generated by d, e and f respectively, where d ∈ D, e ∈ E and f ∈ F. Then (d)r Γ(e)m ρΓ( f )l = DΓEΓF ⊆ P, (d)r ⊆ P or (e)m ⊆ P or ( f )l ⊆ P ⇒ d ∈ P or e ∈ P or f ∈ P. ∴ D ⊆ P or E ⊆ P or F ⊆ P. Hence a bi-interior ideal P is a prime bi- interior ideal of the TGS R. Theorem 3.8. If P1 , P2 , P3 are bi-interior ideals of a TGS R and (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) = P1 ∩ P2 ∩ P3 then every bi-interior ideal of a TGS R is a semi prime BIideal of R. Proof. Let P1 , P2 , P3 be bi-interior ideals of a T GSR&(P1 ΓP2 Γ P3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) = P1 ∩ P2 ∩ P3 . Let P be a bi-interior ideal of a TGS R. To prove, P is a semi-prime ideal of R Suppose P1 ΓP1 ΓP1 ⊆ P1 Then P1 = P1 ∩ P1 ∩ P1 = (P1 ΓP1 ΓP1 ) ∩ (P1 ΓP1 ΓP1 ) ∩ (P1 ΓP1 ΓP1 ) ⊆ P ∩ P ∩ P = P. Hence every bi-interior ideal of R is semi prime. 543 Prime bi-interior Γ-ideals of TG-semring — 544/5 (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 )} ⊆ P1 ΓRΓP1 ∩P1 ΓP1 ΓR∩P1 ΓP1 ΓR ⊆ P1 . It is easy to prove that (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P2 and (P1 ΓP2 ΓP3 )∩(P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P3 . Therefore (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P1 ∩ P2 ∩ P3 ⊆ P. Since P is a strongly prime bi-interior ideal of R, we have P1 ⊆ P or P2. ⊆ P or P3 ⊆ P. Hence P is a strongly irreducible bi-interior ideal of R. Definition 3.9. Let R be a TGS. An element p ∈ R is said to be a regular element if there exists x, y ∈ R and a, b, c, d ∈ Γ such that p = paxbpcydp. Every element in TGS is a regular element then R is known as Regular TGS. Theorem 3.10. R is a regular TG semi ring ⇔ IΓJΓK = In J ∩ K for any left ideal K, lateral ideal J and right ideal I of R. Proof. Let I, J, K be a rt. ideal, medial and a lt. ideal of a regular TGS R respectively. Suppose R be a regular TGS. Clearly IΓJ ΓK ⊆ I ∩ J ∩ K. It enough to show I ∩ J ∩ K ⊆ IΓJΓK. Let p ∈ I ∩J ∩ K. Since R is a regular TG semi ring, there exist a, b, c, d ∈ Γ and x, y ∈ R such that p = paxbpcydp. Since paxbp ∈ I, and pcyd p ∈ J ⇒ paxbpcydp ∈ IΓJΓK. Thus p ∈ IΓJΓK. Hence IΓJΓK = I ∩ J ∩ K. Against suppose that IΓJΓK = I ∩ J ∩ K, for any left ideal K, for any lateral ideal or medial J and right ideal I of R. Let p ∈ R and I be the right ideal generated by p, J be a lateral ideal generated by p and K be a left ideal generated by p. Implies p ∈ I ∩ J ∩ K = IΓJΓK. Since I is a right ideal generated by p, we have p ∈ I implies p = paxbp and also since J is a lateral ideal generated by p, we have p ∈ J implies p = pcyd p. Consider p = paxbp = paxbp .(pcyd p) = paxbpcydp then p is a regular element and thus R is a regular TG semiring. Theorem 3.11. If PΓPΓP = P, for all bi interior ideals P of a TGS R, then TGS R is regular and P1 ∩ P2 ∩ P3 = (P1 ΓP2 ΓP3 ) Γ (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 ) Theorem 3.14. If P is an irreducible bi-interior ideal and PΓP ΓP = P of a regular TGS R then P is a strongly irreducible ideal of R. Proof. Let P1 , P2 and P3 are bi-interior ideals of R such that P1 ∩ P2 ∩ P3 ⊆ P. Then by Theorem [3.11], P1 ∩ P2 ∩ P3 = (P1 ΓP2 ΓP3 ) Γ (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 ) ⇒ (P1 ΓP2 ΓP3 ∩ (P2 Γ P3 ΓP1 )∩ (P3 ΓP1 ΓP2 ) = P1 ∩ P2 ∩ P3 ⊆ P ⇒ P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Thus P is a strongly irreducible ideal of R. Theorem 3.15. Every proper bi-interior ideal P of R is the intersection of all irreducible bi-interior ideals R containing P. Proof. Let R 6= P, BI-interior ideal of T GSR and {Bk : k ∈ ∆} be the collection of irreducible bi- interior ideals and it conT T P ⇒ P ⊆ k∈∆ Bk . Assume that k∈∆ Bk * P. Then tains T ∃p ∈ k∈∆ Bk and p ∈ / P then by the known theorem, ∃Tan irre/ DT ⇒p∈ / k∈∆ Bk ducible bi-interior ideal D∃ P ⊆ D and p ∈ B it is a contradiction. Our assumption that k∈∆ k 6⊂ P is T T wrong. Thus k∈∆ Bk ⊆ P. Hence k∈∆ Bk = P. Theorem 3.16. The intersection of any family of PBIIs of TGS R is a SPBII. for any bi-interior ideals P1 , P2 and P3 of R. Proof. Let PΓPΓP = P, for all BIIs P of a TGS R. Let I be a rt. ideal, J be a medial ideal and L be a lt. ideal of R. Then I ∩ J ∩ L is a BII of R . . . (I ∩ J ∩ L)Γ(I ∩ J ∩ L)Γ(I ∩ J ∩ L)) = (I ∩ J ∩ L) ⇒ I ∩ J ∩ L ⊆ IΓJΓL. We have IΓJ ΓL ⊆ I ∩ J ∩ L. Hence I ∩ J ∩ L = (IΓJΓL) TGS. Theorem 3.12. If P is a BII of R and p ∈ R such that p ∈ P then ∃ an BIIJ of R∃ P ⊆ J and p ∈ J. Theorem 3.13. Suppose R, a regular TGS and PΓPΓP = P, for all BII P of R. Then any BII P of R is STIBII ⇔ P is STPBII. Proof. Given R is a regular Γ -semi ring and PΓPΓP = P, for all BIIs P of a TGS R. Suppose P be a STIBII of R. Now we show that P is a STPBII. Suppose that (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P then by Theorem [3.11], P1 ∩ P2 ∩ P3 = (P1 ΓP2 ΓP3 ) Γ (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 ) for any biinterior ideals P1 , P2 and P3 of R. (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P ⇒ P1 ∩ P2 ∩ P3 ⊆ P ⇒ P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Thus P is a strongly prime bi-interior ideal of R. Reversely assume P is a STPBII of R. Let P1 , P2 and P3 be BIIs of R∃ P1 ∩ P2 ∩ P3 ⊆ P ⇒ (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) = {(P1 ΓP2 ΓP3 ) Γ (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 )} ∩ {(P1 ΓP2 ΓP3 ) Γ (P2 ΓP3 ΓP1 ) Γ (P3 ΓP1 ΓP2 )} ∩{(P1 ΓP2 ΓP3 ) Γ Proof. Let {Bk : k ∈ ∆} be a set of PBIIs of R and P = k∈∆ Bk For any BII P of R, (P1 ΓP2 ΓP3 )∩(P2 ΓP3 ΓP1 )∩(P3 ΓP1 ΓP2 ) ⊆ T k∈∆ Bk = P ⇒ (P1 ΓP2 ΓP3 ) ⊆ (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ Bk , ∀k ∈ ∆ ⇒ (P1 ΓP2 ΓP3 ) ⊆ Bk , ∀ for all k ∈ T ∆ ⇒ (P1 ΓP2 ΓP3 ) ⊆ k∈∆ Bk , k ∈ ∆ = P. Since each Bk are PBIIs, we have P is a PBII of R ⇒ P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Therefore P is a SPBII. Hence the intersection of any family of PBIIs of TGS R is a SPBII. T Remark 3.17. “Family of intersection of BII of R is also a BII of R and it is the set of all BIIs of R form a complete lattice.” Theorem 3.18. Strongly irreducible, semi-prime bi- interior ideal of a TGS R is a strongly prime biinterior ideal. 544 Proof. Let P be a strongly irreducible and semi-prime biinterior ideal of a TGS R. For any bi-interior ideals P1 , P2 and P3 of R, (P1 ΓP2 ΓP3 ) ∩ (P2 ΓP3 ΓP1 ) ∩ (P3 ΓP1 ΓP2 ) ⊆ P. Hence, by Ref.[22, Theorem 22 ] and [35, Theorem 22], P1 ∩P2 ∩P3 biinterior ideal of R. (P1 ∩ P2 ∩ P3 )3 = (P1 ∩ P2 ∩ P3 ) Γ(P2 ∩ P3 ∩ P1 )Γ (P3 ∩ P1 ∩ P2 ) ⊆ (P1 ΓP2 ΓP3 ) and since P is strongly irreducible and semi-prime bi-interior ideal of a TGSR, we have Prime bi-interior Γ-ideals of TG-semring — 545/5 (P1 ∩ P2 ∩ P3 )3 ⊆ (P2 ΓP3 ΓP1 ) , (P1 ∩ P2 ∩ P3 )3 ⊆ (P3 ΓP1 ΓP2 ). Therefore [3] (P1 ∩ P2 ∩ P3 )3 = (P1 ΓP2 ΓP3 )∩(P2 ΓP3 ΓP1 )∩(P3 ΓP1 ΓP2 ) ⊆ P. Since P is a semi-prime bi-interior ideal of R, P1 ∩ P2 ∩ P3 ⊆ P and also since P is a strongly irreducible bi-interior ideal, we have P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Hence P is a strongly prime bi- interior ideal of R. [4] [5] [6] Theorem 3.19. Let R be a TG-semi ring. Prove the following: [7] 1. The family of BI ideals of R is totally ordered set with respect to set inclusion ⇔ [8] [9] 2. Every BII of R is strongly irreducible ⇔ (3) Every BII of R is irreducible. [10] Proof. Given R is a TG-semi ring. Suppose, the set of BI ideals of R is a totally ordered set with respect to ⊆. Now we show that each BII of R is strongly irreducible. Let P be any BI ideal of R. It is enough to show P is a STIBI ideal of R. Let P1 , P2 and P3 be BI ideals of R such that P1 ∩ P2 ∩ P3 ⊆ P. From the hypothesis, we have either P1 ⊆ P2 , P1 ⊆ P3 or P2 ⊆ P3 , P2 ⊆ P1 or P3 ⊆ P1 , P3 ⊆ P2 . . . P1 ∩ P2 ∩ P3 = P1 or P1 ∩ P2 ∩ P3 = P2 or P1 ∩ P2 ∩ P3 = P3 . Hence P1 ⊆ P or P2 ⊆ P or P3 ⊆ P. Thus P is a STIBI ideal of R. Hence (1) ⇒ (2). (2) ⇒ (3) : Let, every BII of R be STI. To show every BI ideal of R is irreducible. Let B be any BII of R ∋ B1 ∩ B2 ∩ B3 = B, ∀BIIsB1 , B2 and B3 of R. Hence from our assumption (2), we have B1 ⊆ B or B2 ⊆ B or B3 ⊆ B. As B ⊆ B1 and B ⊆ B2 and B ⊆ B3 , we have B1 = B or B2 = B or B3 = B. Therefore B is an IBII of R. (2) ⇒ (1): Let each BII of R is an IBI. Let P1 , P2 and P3 be any BIIs of R. Then by the remark, P1 ∩ P2 ∩ P3 is also a BII of R. Hence P1 ∩ P2 ∩ P3 = P1 ∩ P2 ∩ P3 ⇒ P1 = P1∩ P2 ∩ P3 or P2 = P1 ∩ P2 ∩ P3 or P3 = P1 ∩ P2 ∩ P3 by our assumption. P1 ⊆ P2 , P1 ⊆ P3 or P2 ⊆ P3 , P2 ⊆ P1 or P3 ⊆ P1 , P3 ⊆ P2 . The collection of all BIIs of R is a totally ordered set under ⊆. Hence given conditions are equivalent. [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] 4. Conclusion Generalization of ideals of algebraic structures and ordered algebraic structure plays a very remarkable role and also necessary for further advance studies and applications of various algebraic structures. We introduced the notion of prime bi-interior ideal, semi prime bi-interior ideal, irreducible biinterior ideal and strongly prime bi-interior ideal of TGSR and explained axioms and relations between them and also characterized regular TGSR and TGSR using PBI ideals. [2] [22] [23] [24] References [1] [21] P.J. Allen, A fundamental theorem of homomorphism for semi groups, Proc. Amer. Math. Soc., 21(1969), 412–420. Pooth.G.L and Groenewald. 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