S. KAR
Received 31 May 2005
We introduce the notions of quasi-ideal and bi-ideal in ternary semirings and study some properties of these two ideals. We also characterize regular ternary semiring in terms of these two subsystems of ternary semirings.
1. Introduction
Good and Hughes [9] introduced the notion of bi-ideal and Steinfeld [11,12] intro- duced the notion of quasi-ideal. Sioson [10] studied some properties of quasi-ideals of ternary semigroups. In [1], Dixit and Dewan studied about the quasi-ideals and bi-ideals of ternary semigroups. Quasi-ideals are generalization of right ideals, lateral ideals, and left ideals whereas bi-ideals are generalization of quasi-ideals.
In [2], we introduced the notion of ternary semiring. Some work on ternary semiring may be found in [3,4,8,6,7,5].
Our main purpose of this note is to introduce the notions of quasi-ideal and bi-ideal in ternary semirings and study regular ternary semiring in terms of these two subsystems of ternary semirings.
2. Preliminaries
Definition 2.1. A nonempty setStogether with a binary operation, called addition and a ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring ifSis 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 alla,b,c,d,e∈S.
Definition 2.2. LetS be a ternary semiring. If there exists an element 0∈Ssuch that 0 +x=xand 0xy=x0y=xy0=0 for allx,y∈S, then “0” is called the zero element or simply the zero of the ternary semiringS. In this case we say thatSis a ternary semiring with zero.
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:18 (2005) 3015–3023 DOI:10.1155/IJMMS.2005.3015
Throughout this note,Swill always denote a ternary semiring with zero and unless otherwise stated a ternary semiring means a ternary semiring with zero.
Definition 2.3. An additive subsemigroupT ofSis called a ternary subsemiring of Sif t1t2t3∈T, for allt1,t2,t3∈T.
Definition 2.4. An additive subsemigroupIofSis called a left (resp., right, lateral) ideal ofSifs1s2i(resp.,is1s2,s1is2)∈I, for alls1,s2∈Sandi∈I. IfIis both left and right ideal ofS, thenI is called a two-sided ideal ofS. IfIis a left, a right, a lateral ideal ofS, thenI is called an ideal ofS.
An idealIofSis called a proper ideal ifI=S.
Proposition2.5. LetSbe a ternary semiring anda∈S. Then the principal (i)left ideal generated byais given byal= {
risia+na/ri,si∈S;n∈Z0+}, (ii)right ideal generated byais given byar= {
arisi+na/ri,si∈S;n∈Z+0}, (iii)lateral ideal generated byais given byam={
riasi+pjqjarjsj+na/ pj, qj,ri, si∈S;n∈Z0+}, wheredenotes the finite sum andZ0+is the set of all nonnegative integers.
Definition 2.6. A ternary semiring (ring)S is said to be zero divisor free (ZDF) if for a,b,c∈S,abc=0 implies thata=0 orb=0 orc=0.
Definition 2.7. A ternary semiringSis called
(i) multiplicatively left cancellative (MLC) ifabx=abyimplies thatx=y, (ii) multiplicatively right cancellative (MRC) ifxab=yabimplies thatx=y, (iii) multiplicatively laterally cancellative (MLLC) ifaxb=aybimplies thatx=y.
A ternary semiringSis called multiplicatively cancellative (MC) if it is MLC, MRC, and MLLC.
Note 2.8. A multiplicatively cancellative (MC) ternary semiring S is zero divisor free (ZDF).
Definition 2.9[3]. A ternary semiringSwith|S| ≥2 is called a ternary division semiring if for any nonzero elementaofS, there exists a nonzero elementbinSsuch thatabx= bax=xab=xba=xfor allx∈S.
Definition 2.10[2]. An elementain a ternary semiringSis called regular if there exists an elementxinSsuch thataxa=a. A ternary semiring is called regular if all of its elements are regular.
3. Quasi-ideal and bi-ideal in ternary semirings
Definition 3.1. An additive subsemigroupQof a ternary semiringSis called a quasi-ideal ofSifQSS∩(SQS+SSQSS)∩SSQ⊆Q.
Note 3.2. Every quasi-ideal of a ternary semiringSis a ternary subsemiring ofS.
Lemma3.3. Every left, right, and lateral ideal of a ternary semiringSis a quasi-ideal ofS.
Remark 3.4. The converse ofLemma 3.3is not true, in general, that is, a quasi-ideal may not be a left, a right, or a lateral ideal ofS. This follows from the following example.
Example 3.5. LetS=M2(Z−0) be the ternary semiring of the set of all 2×2 square matrices overZ0−, the set of all nonpositive integers. LetQ=
(a0 00) :a∈Z0−
. Then we can easily verify thatQis a quasi-ideal ofS, butQis not a right ideal, a lateral ideal, or a left ideal ofS.
Proposition3.6. IfQis a quasi-ideal of a ternary semiringSandTis a ternary subsemir- ing ofS, thenQ∩Tis a quasi-ideal ofT.
Lemma3.7. The intersection of arbitrary collection of quasi-ideals of a ternary semiringSis a quasi-ideal ofS.
Theorem3.8. An additive subsemigroupQof a ternary semiringSis a quasi-ideal ofSifQ is the intersection of a right ideal, a lateral ideal, and a left ideal ofS.
Proof. LetR be a right ideal,M be a lateral ideal, and Lbe a left ideal of Ssuch that Q=R∩M∩L. Then, by Lemmas3.3and3.7, we find thatQis a quasi-ideal ofS.
The converse ofTheorem 3.8does not hold, in general. But, in particular, we have the following result.
Theorem3.9. An additive subsemigroupQof a ternary semiringSis a minimal quasi-ideal ofSif and only ifQis the intersection of a minimal right ideal, a minimal lateral ideal, and a minimal left ideal ofS.
Proof. LetRbe a minimal right ideal,Ma minimal lateral ideal, andLa minimal left ideal ofSsuch thatQ=R∩M∩L. Then, byTheorem 3.8, it follows thatQis a quasi-ideal ofS.
Now it remains to show thatQis minimal. If possible, letQ⊆Qbe any other quasi-ideal ofS. Then,QSSis a right ideal ofSandQSS⊆QSS⊆RSS⊆R. SinceRis a minimal right ideal ofS, we haveQSS=R. Similarly, we can prove thatSQS+SSQSS=MandSSQ= L. Therefore,Q=R∩M∩L=QSS∩(SQS+SSQSS)∩SSQ⊆Q. Consequently,Q= Qand henceQis a minimal quasi-ideal ofS.
Conversely, letQbe a minimal quasi-ideal ofS. Then,QSS∩(SQS+SSQSS)∩SSQ⊆ Q. Let q∈Q. Then,qSS is a right ideal, (SqS+SSqSS) is a lateral ideal, and SSqis a left ideal ofS. Therefore, byTheorem 3.8,qSS∩(SqS+SSqSS)∩SSqis a quasi-ideal of S, andqSS∩(SqS+SSqSS)∩SSq⊆QSS∩(SQS+SSQSS)∩SSQ⊆Q. SinceQis a min- imal quasi-ideal ofS, we haveqSS∩(SqS+SSqSS)∩SSq=Q. Now it remains to show that qSS, (SqS+SSqSS), and SSqare, respectively, a minimal right, a minimal lateral, and a minimal left ideal ofS. If possible, letRbe any right ideal ofSsuch thatR⊆qSS.
ThenRSS⊆R⊆qSS. Now,RSS∩(SqS+SSqSS)∩SSq⊆qSS∩(SqS+SSqSS)∩SSq=Q.
Thus, by minimality ofQ, we find thatQ=RSS∩(SqS+SSqSS)∩SSq. This implies that Q⊆RSS. Again,qSS⊆QSS⊆(RSS)SS⊆RSS. Thus,qSS=RSS⊆Rand henceR=qSS.
Consequently,qSSis a minimal right ideal ofS. Similarly, we can prove that (SqS+SSqSS) is a minimal lateral ideal andSSqis a minimal left ideal ofS.
Proposition3.10. Any minimal lateral ideal of a ternary semiringSis a minimal ideal ofS.
Proof. LetMbe a minimal lateral ideal ofS. We will show thatMis a minimal ideal ofS.
Letm∈M. Then,SmS+SSmSSis a lateral ideal ofSandSmS+SSmSS⊆SMS+SSMSS⊆ M. SinceM is minimal, we haveM=SmS+SSmSS. Now,MSS=(SmS+SSmSS)SS= (SmS)SS+ (SSmSS)SS⊆SmS+SSmSS⊆M and SSM=SS(SmS+SSmSS)=SS(SmS) + SS(SSmSS)⊆SmS+SSmSS⊆M. This implies thatM is both right ideal and left ideal of S. Consequently,M is an ideal ofS. Now it remains to show thatMis a minimal ideal ofS. If possible, letMbe an ideal ofSsuch thatM⊆M. SinceMis an ideal ofS, it is a lateral ideal ofS. By hypothesis, we haveM=M. Consequently,Mis a minimal ideal
ofS.
Corollary3.11. Any minimal quasi-ideal of a ternary semiringSis contained in a mini- mal ideal ofS.
Proof. LetQbe a minimal quasi-ideal ofS. Then, byTheorem 3.9,Q=R∩M∩L, where Ris a minimal right ideal, M a minimal lateral ideal, andLa minimal left ideal ofS.
Clearly,Q⊆M. FromProposition 3.10, it follows thatMis a minimal ideal ofS.
Proposition3.12. Letxbe an idempotent element of a ternary semiringS, that is,x3(= xxx)=x. IfRis a right ideal,Ma lateral ideal, andLa left ideal ofS, thenRxx,xxMxx, andxxLare quasi-ideals ofS.
Proof. To showRxx,xxMxx, andxxLare quasi-ideals ofS, it is sufficient to show that
Rxx=R∩(SxS+SSxSS)∩SSx, xxMxx=xSS∩M∩SSx, xxL=xSS∩(SxS+SSxSS)∩L.
(3.1)
For the first case, clearly we see thatRxx⊆R∩SSx. Leta∈R∩SSx. Then,a∈Rand a∈SSx. Now, a∈SSximplies that a=n
i=1sitix for somesi,ti∈S. Therefore, axx= (ni=1sitix)xx=n
i=1siti(xxx)=n
i=1sitix=a. Thus, it follows thata∈Rxxand hence Rxx=R∩SSx. Again,a=axx∈SxSand 0∈SSxSS. So we find thata∈(SxS+SSxSS).
Thus,R∩SSx⊆(SxS+SSxSS). Consequently,Rxx=R∩(SxS+SSxSS)∩SSx.
For the second case, We see that xxMxx⊆xSS∩M∩SSx. Leta∈xSS∩M∩SSx.
Then,a∈xSS,a∈M, anda∈SSx. Now,a∈xSSanda∈SSximply thata=m
i=1xsiti= n
j=1ujvjxfor somesi,ti,uj,vj∈S. Therefore, xxaxx=xx
m
i=1
xsiti
xx=
m
i=1
(xxx)siti
xx=
m
i=1
xsiti
xx
=
n
j=1
ujvjx
xx= n j=1
ujvj(xxx)= n j=1
ujvjx=a.
(3.2)
Consequently,a∈xxMxxand hencexxMxx=xSS∩M∩SSx.
The third case can be proved in the same way as in the first case.
We recall the definition of regular ternary semiring.
A ternary semiringSis called regular if for everya∈S, there exists anxinSsuch that axa=a.
Theorem3.13. If, for every quasi-idealQofS,Q3=Q, thenSis a regular ternary semiring.
Proof. IfRis a minimal right ideal,Ma minimal lateral ideal, andLa minimal left ideal ofS, then, byTheorem 3.9, it follows thatR∩M∩Lis a quasi-ideal ofS.
Now, by hypothesis,
R∩M∩L=(R∩M∩L)3
=(R∩M∩L)(R∩M∩L)(R∩M∩L)⊆RML. (3.3)
Again, clearlyRML⊆R∩M∩L. So,R∩M∩L=RMLand hence, by [8, Theorem 3.4],
Sis a regular ternary semiring.
Definition 3.14. A ternary subsemiringBof a ternary semiringSis called a bi-ideal ofSif BSBSB⊆B.
Lemma3.15. Every quasi-ideal of a ternary semiringSis a bi-ideal ofS.
Proof. LetQbe a quasi-ideal ofS. Then we see thatQSQSQ⊆Q(SSS)S⊆QSS,QSQSQ⊆ S(SSS)Q⊆SSQ, and QSQSQ⊆SSQSS. Again {0} ⊆SQS. So, QSQSQ⊆SQS+SSQSS.
Consequently, it follows thatQSQSQ⊆QSS∩(SQS+SSQSS)∩SSQ⊆Qand henceQ
is a bi-ideal ofS.
Note 3.16. The converse ofLemma 3.15does not hold, in general, that is, a bi-ideal of a ternary semiringSmay not be a quasi-ideal ofS.
Remark 3.17. Since every left, right, and lateral ideal ofSis a quasi-ideal ofS, it follows that every left, right, and lateral ideal ofSis a bi-ideal ofS, but the converse is not true, in general.
Proposition3.18. IfBis a bi-ideal of a ternary semiringSandTis a ternary subsemiring ofS, thenB∩Tis a bi-ideal ofT.
Lemma3.19. IfBis a bi-ideal of a ternary semiringSandT1,T2are two ternary subsemir- ings ofS, thenBT1T2,T1BT2, andT1T2Bare bi-ideals ofS.
Corollary3.20. IfB1,B2, andB3are three bi-ideals of a ternary semiringS, thenB1B2B3
is a bi-ideal ofS.
Corollary3.21. If Q1,Q2, andQ3 are three quasi-ideals of a ternary semiring S, then Q1Q2Q3is a bi-ideal ofS.
In general, ifBis a bi-ideal of a ternary semiringSandCis a bi-ideal ofB, thenCis not a bi-ideal ofS. But, in particular, we have the following result.
Theorem3.22. LetBbe a bi-ideal of a ternary semiringS, andCa bi-ideal ofBsuch that C3=C. ThenCis a bi-ideal ofS.
Proof. SinceBis a bi-ideal ofS,BSBSB⊆B, and sinceCis a bi-ideal ofB,CBCBC⊆C.
Therefore,
CSCSC=(CCC)SCS(CCC)
=CC(CSCSC)CC⊆CC(BSBSB)CC⊆CCBCC
=CCBC(CCC)⊆C(CBCBC)C⊆CCC=C.
(3.4) We recall the definition of ternary division semiring.
A ternary semiringSwith|S| ≥2 is called a ternary division semiring if for any nonzero elementaofS, there exists a nonzero elementbinSsuch thatabx=bax=xab=xba=x for allx∈S.
Theorem3.23. A ternary semiringShas no nonzero proper bi-ideals ifSis a ternary divi- sion semiring.
Proof. LetS be a ternary division semiring andB be a nonzero bi-ideal ofS. Let a(= 0)∈B. Then there existss(=0)∈S such thatasx=sax=xas=xsa=xfor allx∈S.
This implies thatS=BSS=SSB. Now,S=BSS=B(SSB)(SSB)=B(BSS)(SBS)(SSB)B⊆ B(BSBSB)B⊆BBB⊆B. Consequently, B=S and hence S has no nonzero proper bi-
ideals.
The converse ofTheorem 3.23is not true, in general. However, in particular, we have the following result.
Theorem3.24. A ternary semiringSis a ternary division semiring ifSis MC and has no nonzero proper bi-ideals.
Proof. LetSbe an MC ternary semiring and has no nonzero proper bi-ideals. Leta(= 0)∈S. Then,aSxandxaSare two bi-ideals ofSfor any nonzerox∈S. SinceSis MC, it is ZDF. So,aSx= {0}andxaS= {0}. By hypothesis, we haveaSx=xaS=Sand hence for x(=0)∈S, there existb,c∈Ssuch thatabx=xac=x. Let ybe any element ofS. Then there existd,e∈Ssuch thatadx=xae=y. Thus,aby=ab(xae)=(abx)ae=xae=y for all y∈S. Now, (yab)ab=y(aba)b=yab. SinceS is MC, we find that yab=yfor all y∈S. Similarly, we can show thatbay=yba=y for all y∈S. Thus, we find that aby=yab=bay=yba=yfor ally∈S, and henceSis a ternary division semiring.
Proposition3.25. LetX,Y, andZbe three ternary subsemirings of a ternary semiringS andB=XYZ. Then,Bis a bi-ideal if at least one ofX,Y,Zis a right, a lateral, or a left ideal ofS.
Proof. LetB=XYZ. SupposeXis a right ideal ofS. Then we find that
(XYZ)S(XYZ)S(XYZ)=X(SSS)(SSS)SSYZ⊆X(SSS)SYZ⊆(XSS)YZ⊆XYZ.
(3.5) Consequently,B=XYZis a bi-ideal ofS.
Now suppose thatY is a right ideal ofS. Then
(XYZ)S(XYZ)S(XYZ)⊆XY(SSS)(SSS)SSZ⊆XY(SSS)SZ⊆XYSSZ⊆XYZ. (3.6) This implies thatB=XYZis a bi-ideal ofS.
Again, ifZis a right ideal ofS, then
(XYZ)S(XYZ)S(XYZ)⊆(XYZ)(SSS)(SSS)SS⊆(XYZ)(SSS)S⊆XY(ZSS)⊆XYZ.
(3.7)
Consequently,B=XYZis a bi-ideal ofS.
Similar proofs can be given for other cases.
Corollary3.26. A ternary subsemiringBofSis a bi-ideal ofSifB=RML, whereRis a right ideal,Mis a lateral ideal, andLis a left ideal ofS.
Proposition3.27. LetBbe a ternary subsemiring of a ternary semiringS. IfRis a right ideal,Mis a lateral ideal, andLis a left ideal ofSsuch thatRML⊆B⊆R∩M∩L, thenB is a bi-ideal ofS.
Proof.
BSBSB⊆(R∩M∩L)S(R∩M∩L)S(R∩M∩L)⊆R(SMS)L⊆RML⊆B. (3.8) The following theorem gives a characterization of a regular ternary semiringSin terms of bi-ideal and quasi-ideal ofS.
Theorem3.28. The following conditions in a ternary semiringSare equivalent:
(i)Sis regular,
(ii)for every bi-idealBofS,BSBSB=B, (iii)for every quasi-idealQofS,QSQSQ=Q.
Proof. (i)⇒(ii). Suppose S is regular. Let B be a bi-ideal of S. Let b∈B. Then there exists x∈Ssuch thata=axa. This implies thata=axaxa∈BSBSB. So we find that B⊆BSBSB. Again, sinceBis a bi-ideal ofS,BSBSB⊆B. Consequently,BSBSB=B.
Clearly, (ii)⇒(iii), by usingLemma 3.15.
(iii)⇒(i). Suppose (iii) holds. LetRbe a right ideal,Ma lateral ideal, andLa left ideal of S. Then,Q=R∩M∩Lis a quasi-ideal ofS, byTheorem 3.8. By hypothesis,QSQSQ=Q.
Now,R∩M∩L=Q=QSQSQ⊆RSMSL⊆RML. Again, clearlyRML⊆R∩M∩L. So, R∩M∩L=RML, and hence, by [8, Theorem 3.4],Sis a regular ternary semiring.
Theorem3.29. A ternary subsemiringBof a regular ternary semiringSis a bi-ideal ofSif and only ifB=BSB.
Proof. IfB=BSB, then it is easy to see thatBis a bi-ideal ofS.
Conversely, suppose thatBis a bi-ideal of a regular ternary semiringS. Letb∈B, then there existsx∈Ssuch thatb=bxb. This implies thatb∈BSBand henceB⊆BSB. Again, BSB⊆BSBSB⊆B. Thus we find thatB=BSB.
Theorem3.30. A ternary subsemiringBof a regular ternary semiringSis a bi-ideal ofSif and only ifBis a quasi-ideal ofS.
Proof. LetSbe a regular ternary semiring. IfBis a quasi-ideal ofS, then, fromLemma 3.15, it follows thatBis a bi-ideal ofS.
Conversely, letBbe a bi-ideal ofS. From [8, Theorem 3.4], we find that ifSis a regular ternary semiring, thenR∩M∩L=RMLfor any right idealR, any lateral idealM, and any left idealL.
Now,
BSS(SBS+SSBSS)SSB
=BSS(SBS+SSBSS)SSB
=B(SSS)B(SSS)B+B(SSS)SB(SSS)SB
⊆BSBSB+BSSBSSB
⊆B+BSB (sinceBis a bi-ideal)
=B+B (byTheorem 3.29)
⊆B.
(3.9)
Consequently,Bis a quasi-ideal ofS.
In view ofLemma 3.19andTheorem 3.30, we have the following result.
Theorem3.31. IfQ1andQ2are two ternary subsemiring andQ3is a bi-ideal of a regular ternary semiringS, thenQ1Q2Q3,Q1Q3Q2, andQ3Q1Q2are quasi-ideals ofS.
In view ofCorollary 3.21andTheorem 3.31, we have the following result.
Corollary3.32. For any three quasi-ideals Q1,Q2,Q3 of a regular ternary semiringS, Q1Q2Q3is a quasi-ideal ofS.
Acknowledgments
The author is grateful to Professor T. K. Dutta for his valuable suggestions and continuous help throughout the preparation of this note. The author is thankful to CSIR, India, for financial assistance.
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S. Kar: Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, India
E-mail address:[email protected]
