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 XYZ⊆U which implies that X⊆U or Y⊆U or Z⊆U 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 X3⊆U which implies that X⊆U for quasi-ideal X∈T.

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)∩TTU⊆U. As U is prime quasi-ideal, UTT⊆U or TUT+TTUTT⊆U or TTU⊆U. Therefore U is a right or latral or left ideal of T.

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

<x>_{q} = {[xTT∩(TxT+TTxTT)∩TTx]+nx : n ϵ Z_{0}^{+}}

THEOREM: 3.5: Let T be a commutative ternary seminear ring. U is a quasi ideal of T. Then U is prime quasi ideal ⇔abc∈U which implies that a∈U or b∈U or c∈U.

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

Then

x= aTT∩TaT+TTaTT∩TTa+nabTT∩TbT+TTbTT∩TTb+nbcTT∩TcT+TTcTT∩TTc+nc
As abc∈U and U be an ideal of T, x∈T. Therefore <a>_{u}<b>_{u}<c>_{u} ⊆ U

As U is a prime quasi ideal of T. Consequently a∈U or b∈U or c∈U.

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 a∈U or b∈U or c∈U.

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)∩TTa⊆U or bTT∩(TbT+TTbTT)∩TTb⊆U or cTT∩(TcT+TTcTT)∩TTc⊆U. If aTT∩(TaT+TTaTT)∩TTa⊆U, then <a>_{u} ⊆ U, which implies that a∈U. bTT∩(TbT+TTbTT)∩TTb⊆U, then <b>_{u} ⊆ U , which implies that b∈U. cTT∩(TcT+TTcTT)∩TTc⊆U, then <c>_{u} ⊆ U, which implies that c∈U. 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 X∩Y∩Z≠ϕ, then X∩Y∩Z⊆XYZ.

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 X∩Y∩Z≠ϕ. Then X∩Y∩Z 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)∩TTU⊆U. 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 UTT≠0. If UTT=0 then U be a non zero right ideal of T satisfying U3=0. Thus contradiction.

If UTT≠0, then UTTUTTU⊆UTT∩(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)∩TTU⊆U.

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'UTTU⊆A'TT∩(TUT+TTUTT)∩TTU=0.

Now A'⊆UTT⇒A'3⊆(A'UTTU)TT=0. By contradiction, 0 be a semiprime ideal of T. Hence A'TT∩(TUT+TTUTT)∩TTU=U, which implies that U⊆A'TT⊆A'. Hence UTT⊆A'TT⊆A'. 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 U⊆TTU∩(TUT+TTUTT)∩UTT⊆U.

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 ATe⊆A∩TTe.

Let x∈A∩TTe. Then x∈A and x∈TTe. Now x∈TTe⇒x=∑i=InSitie ∀ Si,ti∈T.

Consequently,

xee = ∑i=1nSitieee

= ∑i=1nSitieee

= ∑i=1nSitie

= x ⇒x=xee∈Aee⊆ATe

So, A∩TTe⊆ATe. Thus ATe=A∩TTe.

Again, x=xee∈TeT and o∈TTeTT⇒x+o=x∈TeT+TTeTT.

Consequently A∩TTe⊆TeT+TTeTT.

Therefore ATe=A∩(TeT+TTeTT)∩TTe.

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