Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v15i39.1700 research article Prime Quasi-Ideals in Ternary Seminear Rings Vijayakumar R 1 Bharathi A Dhivya dhivyaba.a@gmail.com 2 Assistant Professor, Department of Mathematics, Government Arts College (Autonomous), (Affiliated to Bharathidasan University, Tiruchirapalli) Kumbakonam, Tamil Nadu, 612 002 India Research Scholar, Department of Mathematics,, Government Arts College (Autonomous), (Affiliated to Bharathidasan University, Tiruchirapalli), Kumbakonam, Tamil Nadu, 612 002 India 17 10 2022 15 39 2037 18 8 2022 19 9 2022 2022 Abstract

Objectives/Background: The ternary seminear ring is the generalization of seminear ring and it need not be a ternary semiring. Characterization of quotient ternary seminear rings and some structures of ternary seminear ring have been analysed and also studied ideals in ternary seminear rings. Further quasi ideals in ternary seminear rings defined and discussed about their properties. Methods: Properties of seminear ring and ternary semiring have been employed to carry out this research work to obtain all the characterizations of ternary seminear rings corresponding to that ternary semiring. Findings: We call an algebraic structure (T,+,.) is a ternary seminear ring if (T,+) is a Semigroup, T is a ternary semigroup under ternary multiplication and xy(z+u)=xyx+xyu for all x,y,z,uT. T is said to have an absorbing zero if there exists an element 0T such that x+0=0+x=x for all xT and xy0=x0y=0xy=0 for all x,yT. Throughout this paper T will always stand for ternary seminear ring with an absorbing zero. In this ternary structure we try to study prime quasi ideals concept and obtain their properties. Novelty: In this study, we define the notion of Prime quasi ideals in ternary seminear rings. We also find some of their interesting results.

AMS Subject Classification code: 16Y30,16Y99,17A40

Keywords Ternary seminear ring Idempotent ternary seminear ring Ideals in ternary seminear ring Quasi Ideals in ternary seminear ring Prime Ideals in ternary seminear rings. None
Introduction

Over the centuries, many mathematical theories have been introduced and one such theory is of algebraic structure. Furthermore, various great research was done and is being done by many authors in the area of seminear rings 1, 2, 3, 4. An idea on ternary seminear ring was defined by us5 in 2020 and ideal of ternary seminear ring was defined as T be a ternary seminear ring (T,+,.). A non empty subset I of T is said to be a left(lateral and right) ideal of T if it holds the following conditions i) i+jI for all i,jI ii) t1t2I (respectively t1it2,it1t2I) for all t1,t2T and iI. If I is a left, a lateral and a right ideal of T then I is said to be an ideal of T. Later in 2021, quotient ﻿ternary seminear rings and structures of ternary seminear ring was fathered by us in 6, 7. Then we worked on ideals in ternary seminear rings especially on prime ﻿ideal in ternary seminear rings as T be a ternary seminear ring. A proper ideal I of T is said to be a prime ideal if whenever XYZI then XI or YI or ZI, for all ideals X,Y,Z of T and semiprime ideal in ternary seminear rings as T be a ternary seminear ring. A proper ideal I of T is said to be semiprime ideal if whenever X3I then XI, for any ideals X of T8. Quasi-ideals in ternary seminear rings as T be a ternary seminear ring. Let U be an additive subsemigroup of T. U is said to be a quasi-ideal of T if UTTTUT+TTUTTTTUU and also defined minimal quasi ideals as T be a ternary seminear ring. A non zero quasi ideal U of T is said to be minimal if U does not properly contain any non zero quasi ideal. we proved that an intersection of an arbitrary collection of quasi ideals of T is also a quasi ideal of T 9. In this paper, we define a prime quasi ideals in ternary seminear rings and discuss their properties.

Methodology

In this research work the results of ternary seminear rings such as ideal in ternary seminear rings as prime ideal in ternary seminear rings and semiprime ideal in ternary seminear rings and structures of ternary seminear rings as idempotent in ternary seminear rings and also quasi ideals in ternary seminear ring are used to define the notion of prime quasi-ideals in ternary seminear rings and studied their characterizations.

Results and Discussion

This section deals with Prime Quasi ideals in ternary seminear rings.

DEFINITION: 3.1: Let T be a ternary seminear ring. A proper quasi ideal U of T is said to be prime quasi ideal if XYZU which implies that XU or YU or ZU for quasi-ideals X,Y and Z of T.

DEFINITION: 3.2: Let T be a ternary seminear ring. A proper quasi ideal U of T is said to be semiprime quasi ideal if X3U which implies that XU for quasi-ideal XT.

THEOREM: 3.3: Let T be a ternary seminear ring. U is a quasi ideal of T. If U is prime quasi ideal, then U is a right or lateral or left ideal of T.

Proof. Let T be a ternary seminear ring. U is a prime quasi-ideal of T. Then (UTT)(TUT+TTUTT)(TTU)UTT(TUT+TTUTT)TTUU. As U is prime quasi-ideal, UTTU or TUT+TTUTTU or TTUU. Therefore U is a right or latral or left ideal of T.

DEFINITION: 3.4: Let T be a ternary seminear ring and xT. Then the principal quasi-ideal generated by x is defined as

<x>q = {[xTT∩(TxT+TTxTT)∩TTx]+nx : n ϵ Z0+}

THEOREM: 3.5: Let T be a commutative ternary seminear ring. U is a quasi ideal of T. Then U is prime quasi ideal abcU which implies that aU or bU or cU.

Proof. If U is a prime quasi-ideal of a ternary seminear ring T. abcU. Then U is an ideal of T (from theorem 3.3). Let x ϵ <a>u<b>u<c>u

Then

x= aTTTaT+TTaTTTTa+nabTTTbT+TTbTTTTb+nbcTTTcT+TTcTTTTc+nc

As abcU and U be an ideal of T, xT. Therefore <a>u<b>u<c>u ⊆ U

As U is a prime quasi ideal of T. Consequently aU or bU or cU.

Conversely it is true.

THEOREM: 3.6: Let T be a ternary seminear ring. U is a quasi-ideal of T. Then U is prime quasi [aTT(TaT+TTaTT)TTa][bTT(TbT+TTbTT)TTb][cTT(TcT+TTcTT)TTc]U which implies aU or bU or cU.

Proof. Let T be a ternary seminear ring. U is a prime quasi ideal of T. Let [aTT(TaT+TTaTT)TTa][bTT(TbT+TTbTT)TTb][cTT(TcT+TTcTT)TTc]U for a,b,c in T. Clearly [aTT(TaT+TTaTT)TTa],[bTT(TbT+TTbTT)TTb] and [cTT(TcT+TTcTT)TTc] are quasi-ideals of T. As U is prime quasi-ideal, consequently aTT(TaT+TTaTT)TTaU or bTT(TbT+TTbTT)TTbU or cTT(TcT+TTcTT)TTcU. If aTT(TaT+TTaTT)TTaU, then <a>u ⊆ U, which implies that aU. bTT(TbT+TTbTT)TTbU, then <b>u ⊆ U , which implies that bU. cTT(TcT+TTcTT)TTcU, then <c>u ⊆ U, which implies that cU. Converse is true.

THEOREM: 3.7: Let T be a ternary seminear ring. Then the following criteria are equivalent

i) The quasi-ideals of T are idempotent.

ii) If X,Y,Z are quasi ideals of T such that XYZϕ, then XYZXYZ.

iii) <x>u = [<x>u]3 ∀ x ∈ T

Proof. Let T be a ternary seminear ring,

(i)(ii)

Let X,Y and Z be quasi-ideal of T such that XYZϕ. Then XYZ is a quasi-ideal of T. As each quasi-ideal of T is an idempotent, therefore

X ∩ Y ∩ Z = (X ∩ Y ∩ Z)3

= (X ∩ Y ∩ Z) (X ∩ Y ∩ Z) (X ∩ Y ∩ Z)

⊆ XYZ

(ii)(iii) and (iii)(i)

obviously it is true.

THEOREM: 3.8: Let T be a semiprime ternary seminear ring. Every minimal quasi-ideal U of T is the intersection of a minimal right ideal A, a minimal lateral ideal B and a minimal left ideal C of T.

Proof. Let T be a semiprime ternary seminear ring. As U is a quasi-ideal of T, consequently UTT(TUT+TTUTT)TTUU. U be minimal, so either UTT(TUT+TTUTT)TTU=0 or UTT(TUT+TTUTT)TTU=U.

Suppose UTT(TUT+TTUTT)TTU=0. Then either UTT=0 or UTT0. If UTT=0 then U be a non zero right ideal of T satisfying U3=0. Thus contradiction.

If UTT0, then UTTUTTUUTT(TUT+TTUTT)TTU=0, which implies UTT3 =0 which makes contradiction to our assumption that 0 is a semiprime ideal of T. Hence UTT(TUT+TTUTT)TTU=U. Now, UTT is a minimal right ideal of T.

If there exist a non zero right ideal A' of T such that A'UTT. Then A'TT(TUT+TTUTT)TTU is a quasi ideal of T such that A'TT(TUT+TTUTT)TTUU.

As U is minimal, then either A'TT(TUT+TTUTT)TTU=0 or A'TT(TUT+TTUTT)TTU=U. If A'TT(TUT+TTUTT)TTU=0 .Then A'UTTUA'TT(TUT+TTUTT)TTU=0.

Now A'UTTA'3(A'UTTU)TT=0. By contradiction, 0 be a semiprime ideal of T. Hence A'TT(TUT+TTUTT)TTU=U, which implies that UA'TTA'. Hence UTTA'TTA'. Therefore A'=UTT be a minimal right ideal of T. Similarly, we show that TUT+TTUTT be a minimal lateral ideal of T and TTU be a minimal left ideal of T.

THEOREM: 3.9: Let T be a ternary seminear ring has an identity element one, then each quasi ideal of T is an intersection of a right ideal, lateral ideal and left ideal of T.

Proof. T is a ternary seminear ring with an identity element one. U is a quasi ideal of T. Then TTU is a left ideal, TUT+TTUTT is a lateral ideal and UTT is a right ideal of T. Since T has an identity element I,

<U>a = UTT, <U>b = TUT+TTUTT and

<U>c = TTU

So

U ⊆ <U>a = UTT

U ⊆ <U>b = TUT+TTUTT

and

U ⊆ <U>c = TTU

which implies that UTTU(TUT+TTUTT)UTTU.

Therefore U=TTU(TUT+TTUTT)UTT.

Hence each quasi ideal of T is an intersection of right ideal, lateral ideal and left ideal.

THEOREM: 3.10: Let T be ternary seminear ring. e is an idempotent element of T. If A is a right ideal of T, then ATe is a quasi ideal of T.

Proof. Let T be a ternary seminear ring. e is an idempotent element of T. A is a right ideal of T. To prove ATe is quasi ideal of T. It is enough to prove ATe=A(TeT+TTeTT)TTe. We have ATeATTe.

Let xATTe. Then xA and xTTe. Now xTTex=i=InSitie          Si,tiT.

Consequently,

xee = i=1nSitieee

= i=1nSitieee

= i=1nSitie

= x x=xeeAeeATe

So, ATTeATe. Thus ATe=ATTe.

Again, x=xeeTeT and oTTeTTx+o=xTeT+TTeTT.

Consequently ATTeTeT+TTeTT.

Therefore ATe=A(TeT+TTeTT)TTe.

Hence ATe is a quasi ideal of a ternary seminear ring.

Conclusion

In this research work, prime quasi ideals in ternary seminear rings are defined and their properties have been obtained. The charaterization of quasi ternary seminear ring may be extended to Bi-ideals and Bi-simple ternary seminear rings.

References Koppula K Kedukodi B S Kuncham S P On Prime Strong Ideals of a Seminear Ring Mathematick Vesnik 2020 72 243 256 http://www.vesnik.math.rs/vol/mv20307.pdf Koppula K Kedukodi Babushri Srinivas Kuncham Syam Prasad On perfect ideals of seminearrings Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 2020 62 4 823 842 0138-4821 Springer Science and Business Media LLC https://doi.org/10.1007/S13366-020-00535-2 Radha D Suguna S A Study on Normal Gamma Seminear Ring Journal of Emerging Technologies and Innovative Research (JETIR) 2019 6 2 1075 1081 www.jetir.org Radha D Lakshmi C Raja On Ternary Seminear Rings Journal of Emerging Technologies and Innovative Research 2020 6 2 170 177 www.jetir.org. Vijayakumar R Bharathi Dhivya A On Ternary Seminear Rings International Journal of Mathematics Trends and Technology 2020 66 10 170 177 10.14445/22315373/IJMTT –V66|10P521 Vijayakumar R R Bharathi A Dhivya Quotient ternary seminear rings Malaya Journal of Matematik 2021 9 1 715 719 2319-3786 MKD Publishing House https://doi.org/10.26637/MJM0901/0125 Vijayakumar R Bharathi Dhivya A Structures of Ternary Seminear Rings Stochastic Modellings and Applications 2021 25 2 83 91 https://www.mukpublications.com/resources/smav25-2-10-_vijay_kumar.pdf Vijayakumar . R Bharathi Dhivya A Ideals in ternary seminear rings Proceedings of the National Academy of Sciences 2022 2022 Vijayakumar R Bharathi Dhivya A Quasi-ideals in Ternary Seminear Rings Indian Journal of Natural Sciences 2022