International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 76-84 © TJPRC Pvt. Ltd.,
QUASI-IDEALS AND MINIMAL QUASI-IDEALS IN TERNARY SEMIRINGS 1 1
N. KOTESWARAMMA & 2J. VENKATESWARA RAO
Assistant Professor of Mathematics, Vasireddy Venkatadri Institute of Technology, Nambur, Guntur, Andhra Pradesh, India 2
Professor of Mathematics, Mekelle University, Mekelle, Ethiopia
ABSTRACT This manuscript illustrates quasi-ideals and minimal quasi-ideals in a ternary semiring. Various characterizations of these ideals have been established. In fact the quasi-simple ternary semirings initiated in terms of quasi-ideals and minimal quasi- ideals. It was shown that if any quasi-ideal is itself a quasi-simple ternary semiring, then it becomes a minimal quasi-ideal. Mathematics Subject classification (2010): 16Y60, 16Y99.
KEYWORDS AND PHRASES: Ternary Semiring, Quasi-Ideal, Minimal Quasi-Ideal, Quasi Simple Ternary Semiring.
INTRODUCTION In 1971, Lister (6) introduced the notion of ternary rings and studied some of their properties and the radical theory of such rings. In 2002 Dutta and S.Kar (2) introduced the notion of ternary semiring which is a generalization of ternary ring introduced by Lister. Ternary semiring arises naturally as follows: Consider the ring of integers Z, the subset Z+ of all positive integers of Z is an additive semigroup which is closed under the ring product that is Z+ is a semiring. Now if we consider the subset Z– of all negative integers of Z then we see that Z– is an additive semigroup which is closed under triple ring product; however, Z– is not closed under binary ring product, that is Z– forms a ternary semiring. More generally, in an ordered ring, its positive cone forms a semiring where as its negative cone forms a ternary semiring. Thus a ternary semiring may be considered as a counterpart of semiring in an ordered ring. Steinfeld[9,10,11] introduced first the notion of quasi-ideal for semigroups. Sioson [7] studied some properties of quasi-ideals of ternary semigroups. In [8] Shabir characterized the semirings using quasi-ideals. In [1] Dixit and Dewan studied about the quasi-ideals in ternary semigroups. Quasi-ideals are generalization of right ideals, lateral ideals and left ideals. In [3] S. Kar introduced the notion of quasi-ideal and minimal quasi-ideal in ternary semirings and study some properties of these two ideals which corresponds to those in semiring theory [4,5].
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Quasi-Ideals & Minimal Quasi-Ideals in Ternary Semirings
The main purpose of this note is to study the notions of quasi-ideal and minimal quasi-ideal in ternary semirings and to introduce quasi simple ternary semiring in terms of these two subsystems of ternary semirings.
PRELIMINARIES Definition A nonempty set S together with a binary operation called addition and a ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring if S 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 ∈ S
Example 2.1.1[2]: Let X be a topological space and R– the set of all negative real numbers. Suppose that S = {f: X→R–/f is a continuous map}. We define addition and multiplication on S by (f + g)(x) = f(x) + g(x) and (fgh)(x) = f(x) g(x) h(x) for all x ∈ X and f, g, h ∈ S. Then we can easily check that S forms a ternary semiring. Example 2.1.2[2]: Let Z– be the set of all negative integers. Then with the usual binary addition and ternary multiplication, Z– forms a ternary semiring. Example 2.1.3[3]: Let S be the set of all n × n square matrices over Z–. Then with usual binary addition and ternary multiplication of matrices, S forms a ternary semiring. Definition 2.2[2]: Let S be a ternary semiring. If there exists an element 0 ∈ S such that 0 + x = x and 0xy = x0y = xy0 = 0 for all x, y ∈ S, then “0” is called zero element or simply the zero of the ternary semiring S. In this case we say that S is a ternary semiring with zero. Throughout this note, S will always denote a ternary semiring with zero and unless otherwise stated a ternary semiring means a ternary semiring with zero. Definition 2.3[2]: An additive sub semigroup T of S is called a ternary sub semiring of S if t1t2t3 ∈ T, for all t1,t2,t3 ∈ T. If A, B, C are three subsets of S, then ABC = {Σaibici / ai∈A, bi∈B, ci∈C}.
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Definition 2.4[2]: An additive sub semigroup I of S is called a left (resp., right, lateral) ideal of S if s1s2i (resp. is1s2, s1is2) ∈ I, for all s1, s2 ∈ S and i ∈ I. If I is both left and right ideal of S, then I is called a two-sided ideal of S. If I is a left, a right, a lateral ideal of S, then I is called an ideal of S. An ideal I of S is called a proper ideal if I ≠ S. Proposition 2.5[2]: Let S be a ternary semiring and a ∈ S. Then the principal (i)
Left ideal generated by a is given by <a>l = {Σ risia + na / ri, si ∈ S; n ∈
(ii)
Right ideal generated by a is given by <a>r = {Σ arisi + na / ri, si ∈ S; n ∈
(iii)
Lateral ideal generated by a is given by <a>m = {Σ riasi + Σpjqj arjsj + na / pj, qj, ri, si ∈ S; n ∈
Where Σ denotes the finite sum and
} },
},
is the set of all non negative integers.
Definition 2.6[2]: A ternary semiring S is said to be zero divisor free (ZDF) if for a, b, c ∈ S, abc = 0 implies that a = 0 or b = 0 or c = 0. Definition 2.7[2]: A ternary semiring S is called (i)
Multiplicatively left cancellative (MLC) if abx = aby implies that x = y,
(ii)
Multiplicatively right cancellative (MRC) if xab = yab implies that x = y,
(iii)
Multiplicatively laterally cancellative (MLLC) if axb = ayb implies that x = y.
A ternary semiring S is 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[2]: A ternary semiring S with |S| ≥ 2 is called a ternary division semiring if for any non zero element a of S, there exists a non zero element b in S such that abx = bax = xab = xba = x for all x ∈ S. Definition 2.10[3]: An element a in a ternary semiring S is called regular if there exists an element x in S such that axa = a. A ternary semiring is called regular if all of its elements are regular. Quasi-Ideals We start with the proof of one basic result which we will use quite often. Result 3.1: For each non empty subset x of S the following statements hold (i)
SSX is a left ideal
(ii)
XSS is a right ideal
(iii)
SXS + SSXSS is a lateral ideal of S
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Quasi-Ideals & Minimal Quasi-Ideals in Ternary Semirings
PROOF (i)
SSX =
∑
/
,
Let a,b ∈ SSX. Then a + b = ∑
∑
implies a+b is a finite sum. Hence a + b ∈
SSX and this shows SSX is a sub semigroup of (S,X). For t1,t2∈S, a ∈ SSX, then t1t2a =
∑
=∑
=∑
.
Therefore SSX is a left ideal of S. Similarly we can prove for (ii) and (iii). S.KAR has defined a quasi-ideal Q in a ternary semiring S as follows. Definition 3.2[3]. An additive sub semigroup Q of a ternary semiring S is called a quasi-ideal of S if
. Example 3.2.1[3]: Let S = M2 (
) be the ternary semiring of the set of all 2 x 2 square matrices over
, the set of all non positive integers. Let Q =
0 ": 0 0
$. Then we can easily verify that Q
is a quasi-ideal of S, but Q is not a right ideal, a lateral ideal or a left ideal of S. Note3.3: Let X be a non empty subset of S. By (X)l we mean the left ideal of S generated by X i.e., the intersection of all left ideals of S containing X. Similarly (X)m, (X)r.
PROPERTIES 1.
Every quasi-ideal of a ternary semiring S is a ternary sub semiring of S.
2.
Every left, right and lateral ideal of a ternary semiring S is a quasi-ideal of S. The converse of property 2 is not true, in general, that is, a quasi-ideal may not be a left, a right, or a lateral of S. This follows from the above example(3.2.1).
3.
An additive sub semigroup Q if a ternary semiring S is a quasi-ideal of S if Q is the intersection of a right ideal, a lateral ideal and a left ideal of S.
4.
The sum of two quasi-ideals of S need not be a quasi-ideal of S. We illustrate this by the following example. Example: Let S = M2x2( Q2 =
0 0
0 ":
But Q1 + Q2 = 5.
) be a ternary semiring. and Q1 =
0
0 ": 0
$ and
$ are quasi-ideals of S. 0
0
": ,
$ is not a quasi-ideal of S.
Arbitrary intersection of quasi-ideals of S is a quasi-ideal of S. Proof: Let T = % ∆
/
'(
)* ('_',- . /0
nonempty set T is a subsemigroup of (S,+).
, where ∆ denotes any indexing set, be a
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N.Koteswaramma & J.Venkateswara Rao
Further (SST)
%
(STS+SSTSS)
(TSS) = (SS(% ∆
%
)
%
∆
, 2'
∆
(STS + SSTSS)
Therefore (SST)
%
(TSS)
3
∆
∆.
= T.
Thus arbitrary intersection of quasi-ideals of S is a quasi-ideal of S. 6.
If Q is a quasi-ideal of S, then Q3 = QQQ Proof: As Q is a quasi-ideal of S, SSQ We have Q3 = QQQ
Q. (SQS + SSQSS)
SSQ , Q3 = QQQ
Hence Q3 = SSQ
(SQS + SSQSS)
Thus Q3 = QQQ
Q.
QSS
QSS
QSS and Q3 = QQQ
Q. (SQS + SSQSS).
Q.
7.
If for every quasi-ideal Q of S, Q3 = Q then S is a regular ternary semiring.
8.
If S has an identity element 1, then every quasi-ideal of S is expressed as an intersection of a left ideal, right ideal and lateral ideal of S. Proof: Let S be a ternary semiring with an identity element 1. Let Q be a quasi-ideal of S. Then SSQ is a left ideal, SQS + SSQSS is a lateral ideal and QSS is a right ideal of S. As S contains an identity element 1, we have (Q)l = SSQ , (Q)m = SQS + SSQSS and (Q)r = QSS. Therefore Q
(Q)l = SSQ ,
Q
QSS. But Q
(Q)m = SQS + SSQSS and Q
being a quasi-ideal of S, SSQ SSQSS)
(Q)r = QSS imply Q (SQS + SSQSS)
SSQ
QSS
(SQS + SSQSS)
Q. Therefore Q = SSQ
(SQS +
QSS. Thus every quasi-ideal of S is an intersection of left ideal, lateral ideal and right
ideal. 9.
Intersection of right ideal, lateral ideal and left ideal of S is a quasi-ideal of S. Proof: Let R be a right ideal, M be a lateral ideal and L be a left ideal of S. Then R M L is a sub semigroup of (S,+). Further SS(R M L) SSR
(SMS + SSMSS)
LSS
(S(R M L) + SS(R M LSS)
(R M L) SS
R M L. Hence R M L is a quasi-ideal of S.
MAIN RESULTS Recall that an element e of S is an idempotent if e3 = eee = e. With the help of
idempotent
elements in S we obtain quasi-ideals in S. This we prove in the following theorem. Theorem4.1: Let e be an idempotent element of a ternary semiring S, if R is a right ideal then RSe is a quasi-ideal of S. Proof: To show RSe is a quasi-ideal of S, it is enough if we prove that RSe = R SSe. For that case, clearly we have RSe SSe implies that a =
∑
R
SSe.Let a ∈ R
(SeS + SSeSS)
SSe. Then a ∈ R and a ∈ SSe. Now a ∈
( - for some si,ti ∈ S. Therefore aee =
∑
( - --
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Quasi-Ideals & Minimal Quasi-Ideals in Ternary Semirings
4∑
- -- = ∑
(
Hence RSe = R
( - = a implies that a = aee ∈ Ree
RSe. Therefore R SSe
RSe.
SSe.
Again a = aee ∈ SeS and 0 ∈ SSeSS implies that a+0 = a ∈ SeS + SSeSS. Therefore R SSe
SeS + SSeSS. Consequently RSe = R
(SeS + SSeSS)
SSe.
In the similar way, if M is a lateral ideal of S then eSMSe is a quasi-ideal of S. If L is a left ideal of S then eSL is a quasi-ideal of S. Intersection of a quasi-ideal and a ternary sub semiring of S is a quasi-ideal of that ternary sub semiring of S. We prove this in the following theorem. Theorem 4.2[3]: If Q is a quasi-ideal and T is a ternary subsemiring of S then Q T is a quasi-ideal of T. Proof: As Q
T is a sub semigroup of (S, +) and Q
Further, [TT(Q T)] (QTT)
(SSQ)
T
[T(Q T)T + TT(Q T)TT]
(SQS + SSQSS)
(QSS)
T, we get Q T is a sub semigroup of (T, +). [(Q T)TT]
[TTQ]
[TQT + TTQTT]
Q.
Again [TT(Q T)]
[T(Q T)T + TT(Q T)TT]
Therefore [TT(Q T)] Thus Q
[(Q T)TT]
[T(Q T)T + TT(Q T)TT]
[TTT]
[(Q T)TT]
[TTT + TTTTT]
(TTT)
T.
[Q T] .
T is a quasi-ideal of T.
Definition4.3: A ternary semiring S is said to be a quasi-simple ternary semiring if S is the unique quasiideal of S, that is S has no proper quasi-ideal. A characterization of quasi-simple ternary semiring is furnished in the following theorem. Theorem4.4: If S is a ternary semiring, then S is a quasi-simple ternary semiring if and only if (SSa) (SaS + SSaSS)
(aSS) = S, for all a ∈ S.
Proof: Suppose S is a quasi-simple ternary semiring for any a ∈ S, SSa, SaS + SSaSS, aSS are left, lateral and right ideals of S respectively. Therefore SSa (See property9). Further SSa
S, SaS + SSaSS
S and aSS
S. As S is a quasi-simple ternary semiring. S = (SSa) that S = (SSa) have, S = (SSq)
(SaS + SSaSS)
(SaS + SSaSS)
aSS is a quasi-ideal of S.
S imply SSa
(SaS + SSaSS)
(SaS + SSaSS)
aSS
(aSS). Conversely, Suppose
(aSS). Let Q be quasi-ideal of S. For any q ∈ Q, by assumption we
(SqS + SSq SS)
(qSS)
(SSQ)
S Q. Thus S = Q. Hence S is a quasi-simple ternary semiring.
(SQS + SSQSS)
(QSS)
Q. Therefore
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N.Koteswaramma & J.Venkateswara Rao
Definition 4.5: Let Q be a quasi-ideal of S. Q is said to be minimal quasi-ideal of S if Q does not contain any other proper quasi-ideal of S. Properties of minimal quasi-ideals of a ternary semiring S are proved in the following theorems. Theorem 4.6(refer theorem3.9. of [3]): An additive sub semigroup Q of a ternary semiring S is a minimal quasi-ideal of S if and only if Q is the intersection of a minimal right ideal, a minimal lateral ideal, and a minimal left ideal of S. Proof: Let R be a minimal right ideal, M a minimal lateral ideal and L a minimal left ideal of S such that Q = R M L. Then by property (9), it follows that Q is a quasi-ideal of S. Now it remains to show that Q is minimal. If possible, Let Q1 Q1SS
QSS
RSS
Q be any other quasi-ideal of S. Then Q1SS is a right ideal of S and
R. Similarly we can prove that SQ1S + SSQ1SS = M and SSQ1 = L.
Therefore, Q = R M L = Q1SS
(SQ1S + SSQ1SS)
SSQ1
Q1. Consequently, Q = Q1 and hence Q
is a minimal quasi-ideal of S. (SQS + SSQSS)
Conversely, Let Q be a minimal quasi-ideal of S. Then QSS
SSQ
Q.
Let q ∈ Q. Then qSS is a right ideal, (SqS + SSqSS) is a lateral ideal, and SSq is a left ideal of S. Therefore, by property (9) qSS (SqS + SSqSS) SSq
QSS
(SQS + SSQSS)
quasi-ideal of S, we have qSS
SSQ
SSq is a quasi-ideal of S, and qSS
QSS
(SQS + SSQSS)
SSQ
Q. Since Q is a minimal
(SqS + SSqSS), SSq are respectively, a minimal right, a minimal lateral
and a minimal left ideal of S. If possible, let R be any right ideal of S Such that R R
qSS. Now RSS
(SqS + SSqSS)
of Q, we find that Q = RSS This implies that Q
(SqS + SSqSS)
SSq
(SqS + SSqSS)
RSS. Again, qSS
qSS
(SqS + SSqSS)
qSS. Then RSS
SSq = Q. Thus by minimality
SSq.
QSS
(RSS)SS
RSS. Thus qSS = RSS
R and hence R
= qSS. Consequently, qSS is a minimal right ideal of S. Similarly we can prove that (SqS + SSqSS) is a minimal lateral ideal and SSq is a minimal left ideal of S. Theorem 4.7: If Q is a minimal quasi-ideal of S then any two non zero elements of Q generates the same left (lateral, right) ideal of S. Proof: Let Q be a minimal quasi-ideal of S and x be a non zero element of Q Then (x)l, the left ideal generated by x, is a quasi-ideal of S. Hence (x)l
Q is a quasi-ideal of S. As (x)l
minimal quasi-ideal of S we get (x)l Q = Q. Thus Q implies y∈(x)l . Therefore (y)l
Q
Q and Q is a
(x)l. For any nonzero element y of Q, y∈Q
(x)l. Similarly we can show that (x)l
(y)l . Hence (x)l = (y)l.
In the same way we can prove that any two non zero elements of Q generate the same lateral (right) ideal of S.
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Quasi-Ideals & Minimal Quasi-Ideals in Ternary Semirings
Theorem 4.8: Let Q be a quasi-ideal S. If Q itself is a quasi-simple ternary semiring, then Q is a minimal quasi-ideal of S. Proof: As Q is a quasi-ideal of S, Q is a ternary sub semiring of S (See property 1). Suppose Q is a quasi-simple ternary semiring. Let Q1 be a quasi-ideal of S such that Q1 QQQ1QQ) Q1
(Q1QQ)
SSQ1
(SQ1S + SSQ1SS)
(Q1SS)
Q. Then (QQQ1)
(QQ1Q +
Q1. Therefore Q1 is a quasi-ideal of Q.
Q, Q1 s a quasi-ideal of Q and Q is a quasi-simple ternary semiring imply Q1 = Q. Therefore Q is a
minimalquasi-ideal of S. Proposition 4.9: Any minimal lateral ideal of a ternary semiring S is a minimal ideal of S. Proof: Let M be a minimal lateral ideal of S. We have to show that M is a minimal ideal of S. Let m ∈M. Then SmS + SSmSS is a lateral ideal of S and SmS + SSmSS
SMS + SSMSS
M. Since M is
minimal, we have M = SmS + SSmSS. Now MSS = (SmS + SSmSS) SS = (SmS)SS + (SSmSS) + SSmSS
M and SSM = SS(SmS + SSmSS) = SS(SmS) + SS(SSmSS)
SmS + SSmSS
SmS M. This
implies that M is both right ideal and left ideal of S. Consequently, M is an ideal of S. Now it remains to show that M is a minimal ideal of S. If possible, Let M’ be an ideal of S. By hypothesis, we have M’ = M. Consequently, M is a minimal ideal of S. Corollary 4.10: Any minimal quasi-ideal of a ternary semiring S is contained in a minimal ideal of S. Proof: Let Q be a minima quasi-ideal of S. Then by theorem (4.2), Q = R M L, where R is a minimal right ideal, M a minimal lateral ideal, and L a minimal left ideal of S. Clearly, Q
M. From proposition
4.5, it follows that M is a minimal ideal of S.
CONCLUSIONS This manuscript finally confirms that if S is a ternary semiring, then S is a quasi-simple ternary semiring if and only if (SSa)
(SaS + SSaSS)
(aSS) = S, for all a ∈ S and if Q is a minimal quasi-
ideal of S then any two non zero elements of Q generates the same left (lateral, right) ideal of S. Further it has been acknowledged that any minimal quasi-ideal of a ternary semiring S is contained in a minimal ideal of S.
REFERENCES 1.
V.N.Dixit and S.Dewan, A note on quasi and bi-ideals in ternary semigroups, Int. J. Math. Sci.18(1995), no. 3, 501-508.
2.
Dutta, T.K. and Kar, S (2002): On Reglar Ternary Semirings. Advances in Algebra proceedings of ICM-2002, Satellite conference in Algebra and Related topics at the Chinese University of Hong Kong (August 14-17, 2002) 343.
N.Koteswaramma & J.Venkateswara Rao
3.
84
S.Kar (2005) : On Quasi-ideal and bi-ideals in ternary semirings, International journal of Mathematics and Mathematical Sciences 2005: 18(2005) 3015 â&#x20AC;&#x201C; 3023.
4.
Golan, Jonathan, S (1999) : Semirings and their applications; Kluwer Academic Publishers; Dordrecht.
5.
Hebisch, V. and Weinert, H.J. (1998) : Semirings â&#x20AC;&#x201C; Algebraic Theory and Applications in Computer Science; World Scientific Publishing Company Inc., River Edge; NJ,
6.
Lister, W.G. (1971): Ternary rings; Trans. Amer. Math. Soc., 154, 37.
7.
Sioson, F.M. (1965): Ideal theory in ternary semigroups; Math. Japanica; 10, 63.
8.
Shabir, M., Ali, A., Batool, S., A note on quasi-ideals in semirings. Soucheist Asian Bull. of Math. 27(2004), 923-928.
9.
Steinfeld, O., Quasi-ideals in rings and semigroups. Budapast; Akademiaikiado, 1978.
10. Steinfeld, O., Uber die Quasiideale von Halbgruppen,publ.Math.Debrecen 4 (1956),262-275 (German). 11. Steinfeld, O., Uber die Quasiideale von Ringen,Acta Sci.Math(Szeged) 17 (1956),170-180 (German).