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An Introduction to Ternary 𝛤 −Semirings 

 

G. Chandrasekhar1,2, D.Madhusudana Rao3,P.Siva Prasad4 
1 Research Scholar, Department of Mathematics, Acharya NagarjunaUniversity,Guntur, A. P.,India. 

2Lecturer in Mathematics, Government College (Autonomous), Rajahmundry, A. P., India.  

3Professor of Mathematics, Government College For Women(A), Samba Siva Peta Rd, Opp: AC College, Samba Siva 

Pet, Guntur, Andhra Pradesh, India,mailid:dmrmaths@gmail.com 

4Associate Professor, Department of Computer Science Engineering School of Computing& Informatics, VFSTR 

Deemed to University, Vadlamudi,Guntur,A.P,India, mailid:pusapatisivaprasad@gmail.com 

 

Article History: 

Received: 07-08-2024 

Revised: 17-09-2024 

Accepted: 25-09-2024 

Abstract:  

In this research article has to introduce the concept of ternary Γ-semirings.We first consider 

the congruences and ideals of ternary Γ-semirings then we construct a new ternary Γ-semiring 

and to be discussed formation of ideals on this ternary Γ-semiring. Also with the help of 

congruences induced by homomorphism of a ternary Γ-semiring.After that we will establish 

some of isomorphism theorems and identified the some of commutativity in the diagrams. 

particularly  some fundamental results of ternary Γ-semiring were proved and strengthened. 

Keywords: TernaryΓ-semiring,Ideal,Congruence,Quotient ternary Γ-semiring, 

Homomorphism and Isomorphism 

Mathematics Subject Classification: 16Y60,06B10. 

 

1.Introduction: 

The notion of 𝛤 −semiring was studied by M.K.Rao.as a generalization of 𝛤 −ring as well as of 

semiring. In the year of 1964 𝛤 −ring was introduced by N.Nobusawa There have been a few 

definitions for a 𝛤 −ring.The concepts of ternary 𝛤 −semirings and ternary sub 𝛤 −semiring with 

left,right,lateral was studied by D.Madhusudana Rao and M.Sajani Lavanya  in the year of 2007.T.K 

Datta and M.L Das were introduced and studied the ideals ,prime ideals semiprime ideals k-idelas 

and h-ideals of a ternary 𝛤 −semiring ,regular ternary 𝛤 −semiring respectively 

2..Priliminaries: 

Definition 2.1:Let(𝑇, +) and (Γ, +) be two additive commutative semigroups then 𝑇 is known as 

ternary 𝛤 −semiring if there exist a mapping from 𝑇 × 𝛤 × 𝑇 × 𝛤 × 𝑇  to 𝑇 which maps 

(𝑎, 𝛼, 𝑏, 𝛽, 𝑐) → [𝑎𝛼𝑏𝛽𝑐] satisfying the following conditions 

                        𝑖)[[ 𝑎𝛼𝑏𝛽𝑐]𝛾𝑑𝛿𝑒] = [𝑎𝛼[𝑏𝛽𝑐𝛾𝑑]𝛿𝑒] = [𝑎𝛼𝑏𝛽[𝑐𝛾𝑑𝛿𝑒]] 

 𝑖𝑖)[(𝑎 + 𝑏)𝛼𝑐𝛽𝑑] = [𝑎𝛼𝑐𝛽𝑑] + [𝑏𝛼𝑐𝛽𝑑]     

𝑖𝑖𝑖)[𝑎𝛼(𝑏 + 𝑐)𝛽𝑑] = [𝑎𝛼𝑏𝛽𝑑] + [ 𝑎𝛼𝑐𝛽𝑑] 

                                            𝑖𝑣)[𝑎𝛼𝑏𝛽(𝑐 + 𝑑)] = [𝑎𝛼𝑏𝛽𝑐] + [𝑎𝛼𝑏𝛽𝑑]for all 

𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑇   𝑎𝑛𝑑  𝛼, 𝛽, 𝛾, 𝛿𝜖𝛤 



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Definition 2.2: A ternary𝛤 −semiring 𝑇 is said to have a zero element provided 0 + 𝑥 = 𝑥 = 𝑥 + 0 

and [0𝛼𝑎𝛽𝑏] = [𝑎𝛼0𝛽𝑏] = [𝑎𝛼𝑏𝛽0] = 0, ∀𝑎, 𝑏, 𝑥 ∈ 𝑇 𝑎𝑛𝑑 𝛼, 𝛽𝜖𝛤 

Definition 2.3: A ternary𝛤 −semiring 𝑇 is known as commutative  ternary𝛤 −semiring 𝑇 provided 

[𝑎𝛼𝑏𝛽𝑐] = [𝑏𝛼𝑐𝛽𝑎] = [𝑐𝛼𝑎𝛽𝑏] = [𝑏𝛼𝑎𝛽𝑐] = [𝑐𝛼𝑏𝛽𝑎] = [𝑎𝛼𝑐𝛽𝑏] for all𝑎, 𝑏, 𝑐 ∈ 𝑇.   

Definition 2.4:Let  𝑆 be a non empty subset of a ternary 𝛤 −semiring 𝑇 is said to be a ternary sub  

𝛤 −semiring  of 𝑇 if and only if 𝑆 + 𝑆 ⊆ 𝑆 and [𝑆𝛼𝑆𝛽𝑆] ⊆ 𝑆 for all 𝛼, 𝛽𝜖𝛤.  

Definition 2.5:Let 𝑇 be a ternary𝛤 −semiring and 𝐴 be a non empty subset of 𝑇 is said to be a left 

ternary 𝛤-ideal of 𝑇 if 𝑖)(𝑎 + 𝑏) ∈ 𝐴 and 𝑖𝑖)𝑏, 𝑐 ∈ 𝑇, 𝑎 ∈ 𝐴, 𝛼, 𝛽𝜖𝛤 ⟹ [𝑏𝛼𝑐𝛽𝑎] ∈ 𝐴. 

Definition 2.6:Let𝑇 be a ternary𝛤 −semiring and 𝐴 be a non empty subset of 𝑇 is said to be a lateral 

ternary 𝛤-ideal of 𝑇 if 𝑖)(𝑎 + 𝑏) ∈ 𝐴 and                                                                     𝑖𝑖)𝑏, 𝑐 ∈ 𝑇, 𝑎 ∈

𝐴, 𝛼, 𝛽𝜖𝛤 ⟹ [𝑏𝛼𝑎𝛽𝑐] ∈ 𝐴.     Definition 2.7:Let𝑇 be a ternary𝛤 −semiring and 𝐴 be a non empty 

subset of 𝑇 is said to be a right ternary 𝛤-ideal of 𝑇 if 𝑖)(𝑎 + 𝑏) ∈ 𝐴 and  𝑖𝑖)𝑏, 𝑐 ∈ 𝑇, 𝑎 ∈

𝐴, 𝛼, 𝛽𝜖𝛤 ⟹ [𝑎𝛼𝑏𝛽𝑐] ∈ 𝐴.                      

Definition 2.8:Let 𝑇 be a ternary𝛤 −semiring and 𝐴 be a non empty subset of 𝑇 is said to be a  

ternary 𝛤-ideal of 𝑇 if and only if it is a left ternary 𝛤-ideal, lateral ternary 𝛤-ideal, right ternary 𝛤-

ideal of 𝑇. 

3.Ideals of a ternary  𝛤 −semiring 

Entire of this research article T be a ternary𝛤 −semiringunless otherwise specified. The below 

theorems are easily to prove. 

Lemma3.1:Letbe a non empty index set and { }I  be a family of ideals of (T,𝛤).Then 
I




 is 

an ideal of (T,𝛤). 

Lemma3.2:Let £(T,𝛤)be the set of all ideals of (T,𝛤).Then £ ,Γ), ,( ( , )T    is a complete lattice 

,where I J I J =  and 
I J I J = 

is the unique smallest ideal containing I J . 

Theorem 3.3:LetT be a ternary 𝛤 −semiring with zero and I  be an ideal of                 (T,𝛤).then 

{p / p }
R

I R
I
= + 

is a ternary 𝛤 −semiring with the mapping 
Γ: Γ*

R R R R

I I I I
    →

 defined by 

(p )* *(q )* *(r )I I I p q r I   + + + = +  for all p,q, r ΓR and    

Proof: First we define an operation   on 

R

I  by (p ) (q ) (r )I I Irp qI+ + + =   +  for all 

p I+ , q I+ ,& r I+ 

R

I .It is easy to see that   and * are well defined. Consequently we can verify 

that (

R

I , , [   ] ) is a commutative semigroup and we have the following equalities. 

Left distributive 



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[( ) ( ) (( ) ( )) [( ) ( ) ( )]

[p q (r s)] I

[(p q r) ((p q )]

[[p q r] I] [[p q ] ]

[( ) ( ) ( )] [( ) ( ) ( )]

p I q I r I s I p I q I r s I

s I

s I

p I q I r I p I q I s I

   

 

   

   

   

+ + +  + = + + + +

= + +

=  +

= +  +

= + + +  + + +  

Lateral distributive  

( ) [( ) ( )] (q ) [( ) ( )] ( )]

[p (r s) I] (q I)

[ ( ) ( )]

[ ( ) q ]

[[ ] [ ] ]

[[ ] I] [[ ] ]

[( ) ( ) ( )] [( ) (s ) ( )]

p I r I s I I p I r s I q I

p r s q I

p r s I

p r q p s q I

p r q p s q I

p I r I q I p I I q I

   

 

 

 

   

   

   

+ +  + + = + + + +

= + + +

= + +

= + +

=  +

= +  +

= + + +  + + +  

Right  distributive 

[( ) ( )] ( ) ( )) [( ) ] ( ) ( )

[( ) ]

[( ) ( )]

[[ ] I] [[ ] I]

[( ) ( ) ( )] [( ) ( ) ( )]

r I s I p I q I r s I p I q I

r s p q I

r p q s p q I

r p q s p q

r I p I q I s I p I q I

   

 

   

   

   

+  + + + = + + + +

= + +

=  +

= +  +

= + + +  + + +  

These shows that  is a 

R

I  ternary 𝛤 −semiring. 

Theorem 3.4(correspondance theorem): Let R be a ternary𝛤 −semiring with zero and J an ideal of 

R such that I J  then 

J

I  is an ideal of 
 , ,   ( ) 

R

I


.Conversely ,if K  is an ideal of 
 , ,   ( ) 

R

I


 then 

there exist an ideal J of ( , ,[ ])R  such that I J  and 

J
K

I
=

. 

Proof: The proof of this theorem is similar to the above. 

4.Commutative ternary 𝛤 −semirings and Congruence’s on a  ternary  𝛤 −semirings 

In this entire section R is a  commutative ternary 𝛤 −semiring.The following theorems are well 

known. 

Theorem 4.1: The following conditions on an ideal I of a commutative ternary 𝛤 −semiring R with 

zero are equivalent 

(1) (0 : I) ( : I)H H I =   for all ideals H  of ( , ,[ ])R   



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(2) H I K I =   implies that (0 : ) (0 : )I H I K+ = +  for all ideals H and K of ( , ,[ ])R   where 

( : ) {p / , , }I A R p a b I a b Aand  =       for any .A R   

Theorem 4.2: Assume that R be a commutative ternary 𝛤 −semiring.If I is an ideal of ( , ,[ ])R  ,

.A R   and    then the following statements are holds good: 

(1) ( : ) ( : ) ( : ),I I A I A A A I A A A          

(2) , ( : ) .If A I then I A R =  

Theorem 4.3: Assume that R be a commutative ternary 𝛤 −semiring.If I  is an ideal of ( , ,[ ])R  ,

.A R   then
( : ) ( : \ )

A

If A I I a I A I


 = =
   An equivalence relation   on ( , ,[ ])R   is said to 

be a congruence if for all , , , ,p q r R    we have 

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

p q r p s q s r s

p q p q s q r s r p s and q p s r q s p r s

   

                

 + + +

  

By R:  ,we mean the set of all equivalence classes of the elements of R with respect to the mapping 

  that is, : { ( ) / }.R x x R =   

Lemma 4.4:Let  be a congruence relation on ( , ,[ ])R  .Then

( ) ( ( ) ( )) ( ) ( ( ) ( ) ( )), , , .x y x y and x y z x y z for all x y z R and           + = + =    

Proof:First we observe that ( ) ( ( ) ( ))x y x y   +  +  and ( ) ( ( ) ( ) ( ))x y z x y z       .By 

routine checking, we can easily verify that the above equalities holds. 

In the next theorem, we demonstrate how to construct a new ternary  𝛤 −semiring by using the 

congruence relations. 

Theorem 4.5: Let   be a congruence relation on ( , ,[ ])R  . Define   on R :  by 

( ) ( ) ( ) , .x y x y for all x y R   = +  Then ( : , ,[ ])R    is a ternary  𝛤 −semiring with the 

following mapping : ( : ) ( : ) ( : ) ( : ),R R R R     →  defined by 

( ) (y) (z) ( ), for all x, y, z R, .x x y z        =    

Proof: Let 
' ' '( ) (x ) (y) (y ) (z) (z )x and and     = = = then by lemma 4.4 we have the following 

equality 

' '

' '

( ) (y) (x y) ( (x) (y))

( (x ) (y ))

(x ) (y )

x     

  

 

 = + = +

= +

=   



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Also we have an additional euality

' ' '

' ' '

( ) ( ) (z) (x y z)

( ( ) (y) (z))

( (x ) ( ) ( )

(x ) ( ) ( )

x y

x

y z

y z

       

   

   

    

=

=

=

=  

Thus and  are well defined. 

  Hence,  the author cam verify that  ( : , ,[ ])R    is a commutative ternary semigroup.  Now we 

deduce that 

( ) ( ) [( ( ) ( ))] ( ) ( ) ( )

( ( ))

([ ] [ ])

[ ] [ ]

[ ( ) (y) (z)] [ ( ) (y) (t)]

x y z t x y z t

x y z t

x y z x y t

x y z x y t

x x

        

  

    

     

         

 = +

= +

= +

= 

=   

In that similar way prove that 

[ ( ) [ ( ) ( )] ( ) [ ( ) ( ) ( )] [ ( ) (z) ( )]x y z t x y t x t              = 

[ ( ) ( )] ( ) ( ) [ (x) ( ) ( )] [ (y) ( ) ( )]x y z t z t z t                =   

also 

[[ ( ) (y) ( )] (s) (t)]

[ ( ) [ (y) ( ) (s)] (t)]

[ ( ) (y) [ ( ) (s) (t)]]

x z

x z

x z

        

        

        

=

=  

Therefore  R:  is a ternary  𝛤 −semiring. 

Lemma 4.6: If : :R R R  →  is defined by ( ) ( )R x x =  and RI  is the identity function on  , 

then ( , ) : ( , ) ( : , )R RI R R   →   is an epi-morphism. 

Proof: Let ,x y R and   .Then it is easy to see that  

( ) ( ) ( ) (y)

( ) (y)

R

R R

x y x y x

x

   + = + = 

=   

  Also 

( ) ( ) ( ) (y) (z)

( ) (y) (z)

R

R R R

x y z x y z x

x

          = =

=    

   Clearly R  is surjective  and ( , )R RI  is an epi-morphism. 

 

 



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5 Congruences and products of ternary Γ-semirings 

In this section, we show how to use a ternary Γ-ideal and a congruence on a ternary Γ-semiringRto 

construct a new ternary Γ-ideal of Rand to find the  relationship between them. 

Theorem5.1. Let θ be a congruence on (R, Γ,[ ]).If I is a ternary Γ –ideal of (R, Γ ,[ ]) , then CI={x ∈T| 

[xΓθΓa]∃a ∈I}is a ternary Γ-ideal of    ( R, Γ,[ ]) and  I⊆CI. 

Proof: Clearly  I⊆CI. Let x,y∈CI. Then[xΓθΓa]and [yΓθΓb]for some a,b∈I. On the other hand ,θis a 

congruence on Rwhich implies that                          [(x +y)ΓθΓ(a+b)] and x +y ∈CI. Now, let x∈CI, 

r∈Rand 𝛾∈Γ . Then,  [xΓθΓa]for some a∈I.In other words, θ is a congruence on T which implies that 

[(xΓ𝛾Γr)ΓθΓ(aΓ𝛾Γr)]. Thus ,[x Γ 𝛾Γ r]∈CI. Similarly, we can prove that [r Γ 𝛾Γ x] ∈CI,also we will 

prove that [𝛾 Γ x Γ r] CI,. Therefore, CI is an ternary Γ -ideal of (R, Γ,[ ]). 

By using the standard arguments , we can prove the following theorem. 

Theorem5.2: Let Rbe a ternary Γ -semiring with zero and θ a congruence on (R, Γ,[ ]). Then, θ (0) is 

an ternary 𝛾-ideal of ( R, Γ,[ ]).In the next two theorems, we state the connections between the ideals 

of   (R, Γ,[ ]) and (R: θ, Γ, [ ]). 

Theorem5.3.If I is a ternary Γ -ideal of (R, Γ,[ ]),then I:θ is an ideal of  (R:θ, Γ,[ ]). 

Theorem5.4:  If J is an ternary Γ-ideal of (R:θ, Γ,[ ]), then there exists a ternary                

Γ -ideal I of  (R, Γ,[ ]) such that J=I: θProof:    Define I={x∈R/θ(x)∈J}. Then Wehave 

θ(x) ∈J ⇒x ∈I⇒θ(x) ∈I:θ,andθ(x) ∈I:θ⇒∃a ∈I,θ(x) = θ(a) ⇒θ(x) = θ(a)∈J.Thus, J=I:θNow, suppose 

that x,y∈I. Then θ(x),θ(y)∈J and by Theorem4.5 , we have θ(x+y)=θ(x) ⊕θ(y)∈J. Hence ,x+y∈I. Also 

, assume that x∈I, r∈Rand 𝛾∈Γ .Then, we have θ(x)∈J and by Theorem 4.5, we have θ([x Γ𝛾Γr]) = 

[θ(x)⊙ Γ⊙𝛾⊙ Γ⊙θ(r)] ∈J. Hence , [x Γ𝛾 Γ r] ∈I. Similarly, we can prove that [r Γ𝛾 Γ x] ∈I and [𝛾 

Γ x Γ r] ∈I .Therefore ,Iis an ideal of (R ,Γ,[ ]).  

Lemma5.5:  Let  Ribe a ternary  Γi- semiring(1≤i≤n). Then, R1×···×Rn is a Ternary  Γ 1×···×Γn- 

semiring.The proof is standard and we hence omit the details . It suffices we 

define(x1,···,xn)+(y1,···,yn)= (x1+y1,,xn+yn),and o : (R1×···×Rn)×( Γ 1×···× Γ n)×(R1×···×Rn)−→R1××Rn 

by (x1,···,xn) Γ o Γ (𝛾1,···, 𝛾 n) Γ o Γ (y1,···,yn)=[x1 Γ𝛾1 Γ y1 ],…..,[xn Γ𝛾n Γyn],For all 

(x1,···,xn),(y1,···,yn)∈T1×···×Tnand (𝛾1,···,𝛾n)∈ Γ 1×….× Γ n. 

In the next lemma, we investigate the behavior of congruence on the products of ternary Γ -semirings. 

Lemma 5.6:  Let θi be a congruence on (Ri,Γi,[ ]) for 1≤i≤n .T hen θ is a congruence on (T1×···×Tn , 

Γ 1×···× Γ n,,[ ]) where (a1,···,an) Γ⊙ Γθ Γ⊙Γ (b1…….bn) if 

And only if [aiΓiθiΓi bi]for all ai,bi∈Tiand 1≤i≤n. 

Proof:   If (x1,···,xn) oΓio θΓio(y1,···,yn), then [xi oΓioθioΓioyi]for all 1≤i≤n. Hence (xi+zi) oΓioθi oΓio (yi+zi), 

for all zi∈Tiand 1≤i≤n. This implies that(x1,···,xn)∓(z1,···,zn) oΓioθioΓi(y1,···,yn)∓(z1,···,zn).Also, [xi 

ΓiθiΓiyi]  for all 1≤i≤n implies that [xi oΓio𝛾ioΓiozi]θ[yioΓio𝛾ioΓiozi] for all zi∈Ti, 𝛾i∈𝛾iand 1 ≤ i≤ n. Hence 

[(x1,···,xn)oΓio(𝛾1,···,𝛾n)oΓio(z1,···,zn) ]oΓioθ oΓio [(y1,···,yn) oΓio(𝛾1,···,𝛾n) oΓio(z1,···,zn)].Similarly , we 

can prove that [(z1,···,zn) oΓio(𝛾1,···,𝛾n) oΓio(x1,···,xn)] ]oΓioθ oΓio [(z1,···,zn) oΓio(𝛾1,···,𝛾n) 



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oΓio(y1,···,yn)]Also [(y1,···,yn)oΓio(𝛾1,···,𝛾n) oΓio(x1,···,xn)] ]oΓioθ oΓio [(z1,···,zn) oΓio(𝛾1,···,𝛾n) 

oΓio(y1,···,yn)]Therefore, θ is a congruence on (R1×···×Rn,Γ1×···×Γn,[ ]).  

6       Homomorphism theorems and isomorphism theorems of a ternary Γ –semiring 

In the following theorem , we prove an isomorphism theorem of products of ternary Γ-

semirings. 

Theorem6.1:  Let θibe a congruence on (Ri,Γi,[ ]) for 1≤i≤n and θ the congruence on (R1× ···× Rn, 

Γ1× ···× Γn,[ ]) defined in Lemma 5.6. Then (R1:θ1)×···×(Rn:θn),Γ1×···×Γn 
∼ =(R1×···×Rn:θ, 

Γ1×….×Γn). 

Proof:   By Theorem 4.5 and Lemmas 5.5 and 5.6 , (R1:θ1)×···×(Rn:θn) and R1×···×Rn: θ are 

Γ1×···×Γn-semirings. Define  ψ:(R1:θ1)×···×(Rn:θn)→R1×…..×Rn:θ by: 

ψ (θ1(x1),···,θn(xn)=θ(x1,...,xn)),for all xi∈Ti(1≤i≤n). We can show that (ψ,1Γ1×···×Γn) is an isomorphism 

between ((R1:θ1)×···×(Rn:θn), Γ1×···×Rn,[ ])  and  (R1×···×Rn:θ, Γ1×···×Γn,[ ]). 

We have 

θ1(x1),···,θn(xn)
 ⇐⇒ θ1(y1),··· ,θn(yn) 

⇐⇒ θi(xi)=θi(yi), ∀1≤i≤n 

⇐⇒ xiΓiθiΓiyi, ∀1≤i≤n 

⇐⇒ (x1···xn)ΓiθΓi(y1···yn) 

⇐⇒ θ(x1···xn)=θ(y1···yn) 

⇐⇒ ψθ1(x1),···,θn(xn)=ψθ1(y1),···,θn(yn). 

Hence , (ψ,1Γ1×···×Γn) is well-defined and one to one. Clearly ,(ψ,1Γ1×···×Γn)is onto. Now, we prove that 

(ψ,Γ1×···×Γn) is a homomorphism. We have 

Ψ((θ1(x1),···,θn(xn))∓(θ1(y1),···,θn(yn)))=Ψ(θ1(x1)⊕θ1(y1),···,θn(xn)⊕θn(yn))= 

ψ(θ1(x1+y1),···,θn(xn+yn))= θ((x1 + y1),··· ,(xn+ yn))= 

θ(x1···,xn)⊕θ(y1···,yn)= 

ψ(θ1(x1),···,θn(xn))⊕ψ(θ1(y1),···,θn(yn)). 

Also, we have 

Ψ((θ1(x1),···,θn(xn))o(𝛾1,···,𝛾n)o(θ1(y1),···,θn(yn)))=Ψ(θ1(x1)⊙Γ⊙𝛾1⊙Γ⊙θ1(y1),···,θn(xn)⊙Γ⊙𝛾n⊙Γ

⊙θn(yn))=Ψ(θ1[x1Γ1y1],···,θn[xn𝛾nyn])=θ[x1𝛾1y1,···,xn𝛾nyn)=θ([x1,···xn])⊙Γ⊙(𝛾1,···,𝛾1)⊙Γ⊙θ([y1,···

,yn])= ψ(θ1(x1),···,θn(xn))⊙Γ⊙1𝛾1×···×𝛾n(𝛾1,···,𝛾n)⊙Γ⊙ψ(θ1(y1),···,θn(yn)). 

Therefore, (ψ,1𝛾1×···×𝛾n)is an isomorphism.  

In the next theorems, we consider the congruence on the ternary 𝛾-semiringsT in- duced by the 

homomorphisms and investigate the corresponding results and properties associated with this 

congruence on R. 



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22 

  

Theorem6.2. Let (ϕ,g) :(R1,Γ1,[ ])−→(R2,Γ2,[ ])be a homomorphism . Define the relation θ(ϕ,g) on 

(R1,Γ1) as follows:[xΓ1θ(ϕ,g)Γ1y]⇐⇒ϕ(x)=ϕ(y). 

Then θ(ϕ,g) is a congruence on (R1Γ1,[ ]). 

Proof.  Clearly ,θ(ϕ,g) is an equivalence relation. Suppose that xθ(ϕ,g)y. We have 

ϕ(x)=ϕ(y) =⇒ϕ(x)+ϕ(z)= ϕ(y)+ϕ(z) =⇒ϕ(x+z)= ϕ(y+z) for all z∈R1.Thus [(x+z)Γ1θ(ϕ,g)Γ1(y+z)]. Also, 

we have 

ϕ(x)=ϕ(y)=⇒[ϕ(x)Γ1g(𝛾)Γ1ϕ(z)]=[ϕ(y)Γ1g(𝛾)Γ1ϕ(z)]=⇒ϕ([xΓ1𝛾Γ1z])=ϕ([yΓ1𝛾Γ1z]) 

for all z∈R1 and 𝛾∈Γ1. Therefore ,θ(ϕ,g) is a congruence on(R1,Γ1,[ ]). 

Theorem6.3: Let (ϕ,g):(R1Γ1,[ ])−→(R2,Γ2,[ ]) be a homomorphism. Set A= {I⊆R1|θ(ϕ,g)⊆I×I}and B= 

{J|J⊆R2}.Then, there exists an 1-1 mapping from A to B. 

Proof .Define ψ :A→B by ψ(I)=ϕ(I). Clearly ,ψ is well-defined. 

Suppose that ψ(I1)=ψ(I2).Then ϕ(I1)=ϕ(I2).Also we can see that 

x∈I1 =⇒ ϕ(x) ∈ϕ(I1)=ϕ(I2) =⇒ ∃y∈I2,ϕ(x)=ϕ(y) 

=⇒ (x,y)∈θ(ϕ,g)⊆I2×I2 

=⇒ x∈I2 

=⇒ I1⊆I2.Similarly,I2⊆I1andsoI1=I2and hence, ψisone-to-one. 

Theorem6.4.Let (R1,Γ1,[ ])
(ϕ

1
,g

1(R2,Γ2,[ ])
(ϕ

2
,g

2
)(R3,Γ3,[ ])be a sequence of 

Homomorphisms .Then (ψ,g):(R1×R2×R3,Γ1×Γ2×Γ3,[ ])→(R1×R2×R3,Γ1×Γ2×Γ3,[ ]) 

Defined by ψ(x,y)=ϕ1(x),ϕ1(y) and g(𝛾,β)=g1(𝛾),g1(β) for allx, y∈R1  and 𝛾,β∈Γ1, is a homomorphism 

such that ψ(θ(ϕ1,g1))⊆θ(ϕ2,g2). More- over , if (ψ1,g1) is on to and (ψ2,g2) is one to one ,then ψ( 

θ(ϕ1,g1))=θ(ϕ2,g2). 

Proof.  It is trivial that (ψ,g) is a homomorphism.Hence,we have 

ψ(a,b)∈ψ(θ(ϕ1,g1)),(a,b)∈θ(ϕ1,g1) =⇒ ψ1(a)=ψ1(b) 

=⇒ ψ2(ψ1(a)=ψ2(ψ 1(b)) 

=⇒ (ψ1(a),ψ1(b))∈θ(ψ2,g2) 

=⇒ ψ(a,b)∈θ(ψ2,g2). 

Thus, ψ(θ(ψ1,g1))⊆θ(ψ2,g2).Now, if (ψ1,g1) is surjective and (ψ2,g2) is                         in-jective, then 

we have ψ( θ(ψ1,g1))=θ(ψ2,g2)It suffices to prove that θ(ψ2,g2)⊆ψ( θ(ϕ1,g1)) Hence, we have 

(t,t')∈θ(ψ2,g2)=⇒ ψ2(t)= ψ2(t
') 

=⇒ ∃a,b∈R,ψ1(a)=t,ψ1(b)=t' 

=⇒ (t,t')= ψ(a,b)=(ψ1(a),ψ1(b)) 

=⇒ (t,t')∈ψ( θ(ψ1,g1)) 



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This shows that θ(ϕ2,g2)⊆ψθ(ϕ1,g1), and the proof is completed.  

Theorem6.5..Let (R1,Γ1,[ ])
(ϕ

1
,g

1(R2,Γ2,[ ])
(ϕ

2
,g

2
)(R3,Γ3,[ ])be a sequence of 

Homomorphism’s. Then Imψ1×Imψ1⊆θ(ψ2,g2) if and only if ψ2 o  ψ1 is constant. 

Proof. The proof of the necessary part is routine and we only prove the sufficiency part. 

(=⇒):Letx,y∈R1.Then, (ψ1(x),ψ1(y))∈Imψ1×Imψ1⊆θ(ψ2,g2). Hence, 

ψ2ψ1(x)=ψ2ψ1(y).This show that ψ2oψ1is a constant.  

Finally, by the congruence on the ternary 𝛾-semiring induced by homomorphism , we are able to 

establish some isomorphism theorems and investigate the commu- tativity of some diagrams. 

Theorem6.6. (IsomorphismTheorem) If (ψ,g):(R1, Γ1,[ ])−→(R2,Γ2,[ ]) is an epimorphism,then there 

exists an unique isomorphism       

(ψ,g):(R1:θ(ϕ,g),Γ1,[ ])−→(R2,Γ2,[ ]) Such that the following diagram commutes:                                                                                                                                                                                                                                                            

 

(ψ,g) 

 

    (R1,Γ1,[     ])         (R1,Γ2,[     ]) 

      

                            

                                  (𝛱R1,1R1)  

       (ψ,g) 

   

 

(R1,:θ(ψ,g),Γ1,)) 

Where ΠR1,:R1,→R1,: θ(ϕ,g) is defined by ΠR1(x)=θ(ϕ,g) (x) for all x∈R1, 

And1R1is identity. 

Proof. Define ψ:𝑅1:θ(ϕ,g)−R2 by ψ(θ(ϕ,g)(x))=ψ(x) for all x∈R1.Then we haveθ(ϕ,g)(x)= 

θ(ϕ,g)(y)⇐⇒xθ(ϕ,g)y⇐⇒ψ(x)=ψ(y),and hence ψ is well defined and is a 1-1 mapping.Now,(ψ,g) is a  

homomorphism. We have 

ψ(θ(ϕ,g)(x)⊕θ(ϕ,g)(y))=ψ(θ(ϕ,g)(x+y)) 

=ψ(x+y)=ψ(x)+ψ(y) 

=ψ(θ(ϕ,g)(x))+ψ(θ(ϕ,g)(y)). 

We deduce that  Ψ(θ(ϕ,g)(x)⊙Γ⊙𝛾⊙Γ⊙θ(ϕ,g)(y)=ψθ(ϕ,g)(x⊙Γ⊙𝛾⊙Γ⊙y) 

=ϕ(x⊙Γ⊙𝛾⊙Γ⊙y)=ϕ(x)⊙Γ⊙g(𝛾)⊙Γ⊙ϕ(y)=ψ(θ(ϕ,g)(x))⊙Γ⊙g(𝛾)⊙Γ⊙ψ(θ(ϕ,g)(y)). 



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=θ(ϕ2,g2)ϕ1(x)⊕θ(ϕ2,g2)ϕ1(y) 

1, 1 1, 1( (( ( )) ( ( )g gx y    =   

 

 

 

 

 

 

θ(ϕ1,g1)(x)⊙g1(γ)⊙ψθ(ϕ1,g1)(y). 

Therefore,(ψ,g) is a homomorphism . Also ϕ(x) = ψθ(ϕ,g)(x)= ψΠR1(x) and      go1𝛾1 = g which imply 

that the diagram is commutative . Let 

(ψ,g):(R1:θ(ϕ,g), Γ1,[ ])−→(R2, Γ2,[ ]) 

Be such that ψoΠR1=ϕ.Then, Wehave 

 

ψθ(ϕ,g)(x)=ψΠR1(x)=ϕ(x)=ψΠR1(x)=ψθ(ϕ,g)(x). 

Thus ,(ψ,g) is unique and the proof is completed.  

Theorem 6.7. Let (R1 ,Γ1,[ ]) (ϕ1,g1),  (R2,𝛾2,[ ])(ϕ2,g2)(R3,𝛾3,[ ]) be a  

Sequence of Homomorphism’s .Then ,there exists an unique homomorphism 

(ψ,g1):(T1:θ(ϕ1,g1),𝛾1)→(T2:θ(ϕ2,g2),𝛾2) 

Such that the following diagram is commutative: 

(ϕ1,g1) 

(R1,𝛾1)  (R2,𝛾2) 

   

(ΠR1, 1R1 )  (ΠR2, 1R2 ) 

   

(R1:θ(ϕ1,g1),𝛾1,[ ])  (R2:θ(ϕ2,g2),𝛾2,[ ]) 

Moreover ,if(ϕ1,g1) is on to and (ϕ2,g2) is 1-1,then(ψ,g1) is an isomorphism. 

Proof .Define ψ: R1: θ(ϕ1,g1)R2: θ(ϕ2,g2)by ψ(θ(ϕ1,g1)(x))= θ(ϕ2,g2)ϕ1(x). Then, ψ is well-

defined.Finally,we need prove that(ψ1,g1) is a homomorphism. We have 

Ψ(θ(ϕ1,g1)(x)⊕θ(ϕ1,g1)(y))=ψ(θ(ϕ1,g1)(x+y))=θ(ϕ2,g2)ϕ1(x+y)=θ(ϕ2,g2)ϕ1(x)+ϕ1(y) 

 

 

Also, we have 

Ψ(θ(ϕ1,g1)(x)⊙Γ⊙𝛾⊙Γ⊙θ(ϕ1,g1)(y))=ψ(θ(ϕ1,g1)(x⊙Γ⊙y) 

=θ(ϕ2,g2)ϕ1(x⊙Γ⊙y) 

=θ(ϕ2,g2)ϕ1(x)g1(𝛾)ϕ1(y) 

=θ(ϕ2,g2)ϕ1(x)⊙g1(𝛾)⊙θ(ϕ2,g2)ϕ1(y) 

= 

Therefore, (ψ,g1) is a homomorphism .Also ,we have 



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ψΠR1(x)= ψθ(ϕ1,g1)(x)= θ(ϕ2,g2)ϕ1(x)= ΠR2ϕ1(x), and  g1o1𝛾1=1𝛾2og1.This shows that the diagram is 

commutative. 

Let(ψ¯,g1):(T1:θ(ϕ1,g1),𝛾1,[ ])−→(T2:θ(ϕ2,g2),𝛾2,[ ]) be a homomorphism which 

Makes the diagram commutative.  Then, we have 

 

ψ¯θ(ϕ1,g1)(x)= ψ¯ΠT1(x)=Πt2ϕ1(x)=θ(ϕ2,g2)ϕ1(x)=ψθ(ϕ1,g1)(x). 

Thus, (ψ,g1) is unique and the proof is completed. 

Acknowledgements: 

The first and second authors express their warmest thanks to the research director Dr. D. Madhusudana 

Rao, Department of Mathematics, Govt. Women degree college, Sambasivapet, Guntur.  The authors 

would like to thank the experts who have contributed towards preparation and development of this 

research article and the referees, Chief Editor for the valuable suggestions and corrections for the 

improvement of this research article. 

REFERENCES  

[1] Dutta. T. K. and Kar. S.,On Regular Ternary Semiring, Advances in Algebra, Proceedings of the ICM Satellite 

Conference in Algebra and Related Topics, World Scientific (2003), 343-355. 2.  

[2] Dutta. T. K and Das. M. L., On Strongly PrimeSemiring, Bull. Malays. Math. Sci. Soc. (2) 30 (2) (2007), 135- 141. 

[3] Handelman. D. and Lawrence. J.,Strongly Prime Rings, Trans. Amer. Mat. Soc. 211 (1975), 209-223. 

[4] Lister, W. G.,Ternary Rings, Trans. Amer. Math. Soc. 154 (1971), 37-55. 5.  

[5] SajaniLavanya. M, Madhusudhana Rao. D. and Syam Julius Rajendra. V.,On Lateral Ternary Γ-Ideals of Ternary Γ-

Semirings, American International Journal of Research in Science, Technology, Engineering & Mathematics, 12(1), 

September-November, 2015, 11-14. 

[6] SajaniLavanya. M, Madhusudhana Rao. D. and Syam Julius Rajendra. V., On Quasi-Ternary -Ideals and BiTernary 

-Ideals In Ternary -Semirings, International Journal of Mathematics and Statistics Invention (IJMSI), Volume 3 Issue 

6, (September.2015), PP-05-14. 

[7] SajaniLavanya. M, Madhusudhana Rao. D. and Syam Julius Rajendra. V.,Prime Bi-Ternary -Ideals in Ternary 

Semirings, British Journal of Research, (2), (6), (2015) 156-166. 

[8] SajaniLavanya. M, Madhusudhana Rao. D. and Syam Julius Rajendra. V.,A Study on The Jacobson Radical of A 

Ternary -Semiring, International Journal of Mathematics and Computer Applications Research (IJMCAR), Vol. 6, 

Issue1,(Feb 2016), 17-30. 

[9] P. A. Grillet,Semigroups, an introduction to structure theory, Dekker New York 1995.  

[10] M. P. Grillet, P. A. Grillet, Completely 0-simple semirings, Trans. Amer. Math. Soc. 155 (1971) 19-33.  

[11] H. Hedayati,Generalized fuzzy k-ideals of semirings with interval-valued membership functions, Bull. Malays. Math. 

Sci. Soc. 2(32)3 (2009) 409-424.  

[12] H. Hedayati, R. Ameri, Regular and fundamental relations on k-hyper ideals of semi-hyper rings, Bull. Calcutta Math. 

Soc. 101(2) (2009) 105-114.  

[13] H. Hedayati, B. Davvaz, Fundamental relation on Γ-hyper rings, Accepted by Arts Combinatoria.  

[14] S. Kar, Ideal theory in the ternary semiring Z− 0, Tsukuba J. Math. 11(2) (1987) 371-382. 

[15] S. Kyuno, Prime ideals in gamma rings, Pacific J. Math. 98(2) (1982) 375-379. 

[16] G. Srinivasa Rao P. Siva Prasad, M. Vasantha, Dr. D. Madhusudhana Rao “On Strongly Duo and Duo Left Γ-TS-Acts 

over ternary-semigroups”, International Journal of Pure and Applied  Mathematics Volume 113 No. 6 2017, 65 – 73, 

ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) PP 67-73. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

592 https://internationalpubls.com 

[17] Ch. ManikyaRao,  P. Siva Prasad, D. Madhusudhana Rao, G. SrinivasaRao,”Maximal ideal of compact connected 

topological ternary semigroups” International conference on mathematics 2015,At: kerela,Volume: Volume 4 Issue 

2 

[18] P. Sivaprasad , Dr. D. Madhusudhana Rao , G. Srinivasa Rao,” A STUDY ON STRUCTURE OF PO-TERNARY 

SEMIRINGS”, JOURNAL OF ADVANCES IN MATHEMATICS, Vol .10, No.8,PP:3717-3724. 

[19] P. Sivaprasad , Dr. D. Madhusudhana Rao , Mamidipalli. Vasantha, , B. Srinivasa Kumar,” On Γ-TS-Acts Over 

Ternary Γ-Semigroups” International Journal of Engineering & Technology, 7 (4.10) (2018)PP:812-815. 

[20] G. Srinivasa Rao, D. Madhusudhanarao and P. Siva Prasad, Simple Ternary Semi-rings, The Global Journal of 

Mathematics & Mathematical Sciences, 9(2) (2016), 185-196.  

[21] D. Madhusudhana Rao, G. Srinivasa Rao, Special Elements in ternary semi rings, International Journal of Engineering 

Research and Applications, 4(11) (2014), 123-130.  

[22] G. Srinivasa Rao, D. Madhusudhana Rao, Structure of certain ideals in ternary semi rings, Int. J. of Innovative Science 

and Modern Engg., 3(3) (2015), 49-56. 

[23] G. Srinivasa Rao, D. Madhusudhana Rao, A Study on Ternary Semi rings, Int. J. of Math. Archive, 5(12) (2014), 24-

30.  

[24] G. Srinivasa Rao, D. Madhusudhana Rao, Characteristics of Ternary Semi rings, Int.J. of Engg. Res. and Mgt., 2(1) 

(2015), 3-6.  

[25] G. Srinivasa Rao, A. Nagamalleswara Rao, P.L.N. Varma, D.Madhusudhana Rao, Ch.   Ramprasad, Prime Bi-interior 

ideals in TGSR, Malaya Journal of Mathematika, Vol.9, No.1, pp:542-546, 2021. 

[26] G. Srinivasa Rao, A. Nagamalleswara Rao, P.L.N. Varma, D. Madhusudhana Rao, Ch. Ramprasad, Bi-interior ideals 

in TGSR, Advances in Mathematics Scientific Journal, 10 (2021), No.3, pp: 1183-1195. 

[27] Siva Prasad.P, Revathi. K, 2, Sundarayya. P , Madhusudhana Rao. D , “Compositions of Fuzzy T-Ideals in Ternary -

Semi ring”, International Journal of Advanced in Management, Technology and Engineering Sciences Volume 7, 

Issue 12, 2017 ISSN NO : 2249-7455,PP:135-145. 

[28] P. Sivaprasad, C. Sreemannarayana, D. Madhusudhana Rao, T. Nageswara Rao, K. Anuradha,” On Le- Ternary Semi 

groups-I” International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-7, Issue-

ICETESM, March 2019, PP:165-167. 

[29] P. Sivaprasad, C. Sreemannarayana, D. Madhusudhana Rao, T. Nageswara Rao, Sajani Lavanya. M,” On Le- Ternary 

Semi groups-II” International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-7, 

Issue-ICETESM, March 2019, PP:168-170.