Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 222 https://internationalpubls.com On Primitive Ternary Г-Semirings K. Maha lakshmi,1,2, P. Siva Prasad3, D.Madhusudana Rao4 1 Research Scholar, Department of Mathematics, VFSTR Deemed to be University, Vadlamudi, Guntur, Andhra Pradesh, India, mailid: mhlakahmi@gmail.com. 2Assistant Professor, Department of Mathematics, Vignan Nirulla Engineering College, Guntur, Andhra Pradesh, India, mail id: mhlakahmi@gmail.com. 3Associate Professor, Department of Computer Science & Engineering, School of Computing & Informatics, VFSTR Deemed to University, Vadlamudi, Guntur,A.P, India, mailid:pusapatisivaprasad@gmail.com 4Professor 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 Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: Subsequent to presenting the thoughts of primitive Ternary Г-semi ring along with primitive ideal of a Г-semi ring we concentrate on them through operator semi ring and get a few outcomes similar to those of semi ring hypothesis. Keywords: Semi-irreducible, Irreducible, faithul ГS-semimodules, primitive Ternary Г-semi ring. AMS Mathematics Subject Classification. 16Y60, 16Y99, 20N10 1. Introduction : Presenting the thought of Ternary Г-semiring S-Semimodule entitled ГS-semimodule alongside the thoughts of semi-irreducible, irreducible, faithul ГS-semimodules with a goal to present the idea of primitive Г-semiring in addition to prospect to present the idea of Jacobson radical. Now we concentrate on primitive TГ-semiring by means of the operator semirings. We confirm that by [6] a right operator semiring R is primitive iff TГ-semiring S is primitive. In conclusion, “primitive h-ideal of a TГ-semiring S utilizing the connection amid the Annihilator of an irreducible TГS-semimodule M in S with the aim of M in the right operator semiring R of TГ-semiring S”. “Throughout this paper TГ-semiring - TГ-S,Ternary ГS-semimodule denoted by TГS-S, Ternary ГS- subsemimodule denoted by TГS-SS, irreducible TГ-semimodule- ITГ-S, irreducible TГS- semimodule- ITГS-S , Primitive Ideal-PI, right operator semiring-ROS faithful irreducible TГS-semimodule- FTГS-S, faithful irreducible R-semimodule-FIR-S, irreducible R-semimodule-IR-S, faithful irreducible / *R P  -semimodule- FI ( / * )R P  -S, additive commutative monoid-ACM, TГS-semiring- TГSS” Preliminaries Definition 2.1: Let U,  be two additive commutative semigroups. Then U is entitled a ternary gamma semiring presented  a mapping U U U U  → suiting the next conditions: (1) ( ) ( ) ( )x y z p q x y z p q x y z p q           = = (2) [( ) ] [( ] [ ]p q r s p r s q r s     + = + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 223 https://internationalpubls.com (3) [ ( ) ] [( ] [ ]p q r s p q s p r s     + = + (4) [ ( )] [( ] [ ] , , , & , , , .p q r s p q r p q s p q r s U         + = +    An ideal J in a TГ-semiring S is called a k-ideal if 1 1 1 1 1, , .x y J x S y J x J+      In a TГ-S semiring S an ideal I is entitled an ‘h-ideal’ if 1 1 1 2 1 1 1 1 2 1, , & , .x y z y z x z S y y I x I+ + = +     permit S be a TГ-S & free additive commutative semigroup G generate by .S S  subsequently relation  on G, defined by If [ , ] [ , ]i i j j i j k l =  in R then , .i i i j j j i j s t k s t k s t S   =    Semi-irreducible, Irreducible, Faithful TГ-Semimodules : Definition 3.1: Permit S is a TГ-S. An ACM ‘M’ is known as merely TГS-S, if N S S N   → (Images to be denoted by : , ,k s t k N     ) satisfying the following conditions: (i) ( )k l s t k s t l s t     + = + (ii) ( )k s t u k s t k s u     + = + (iii) ( )k s t u k s u k t u     + = + (iv) ( ) ( ) ( )k s t u v k s t u v k s t u v           = + (v) 0 0 0 0 , , , , , .N N S St u k s k s k l N s t u v S     = = =    In addition to the above conditions if ,k e f k k N  =   where {e, f}is an identity element of S, then N is believed to be a unitary right ternary ГS-semimodule. A left TГS-semimodule can be defined likewisely. Example: 3.2: A TГSS ‘S’, wherever the ‘additive commutative semigroup’ S of all 2 3 matrices above 0 &Q +  is also ‘additive commutative semigroup’ of all 3 2 matrices over the identical set & n k l  denote product of matrices of , , , , ; , , & ,n k l n N k l S      now the right unity of S is 3 1 [ , , , ]i i i i i e f  =  where 1 1 2 2 3 3 1 1 2 2 3 3 1 0 0 0 0 0 0 1 , 0 1 , 1 0 , 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 , , .31 1 0 0 0 0 0 0 13 3 f e f e f e                   = = = = = =                             = = = = = =                  A subset ( )N  of a TГS-S N is known as a TГS-S of N if (i) k l N+  , (ii) , , , , ,k s t N s t S k l N        holds the zero of N. A TГS-SS N of a TГS-subsemimodule O is named KГS-subsemimodule O if , , .k l N l N k O k N+      Let N be a TГS-SS of a TГS-subsemimodule O. Subsequently k- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 224 https://internationalpubls.com closure of N, is described by { : : , }.N j O j k l k l N=  + =  A TГS-SS ‘N’ of a TГS-S O is named as ternary (hГS-) subsemimodule of O if 1 1 1 2 1 1 2 1 1 1, , , , .x n z n z n n N x z O x N+ + = +     Let N be a TГS-SS of a TГS-S ‘O’. Afterward h-closure of N, is named by N = 1 2 1 2{ : , , , }.o N o n z n z n n N z O + + = +   Proposition 3.03: Let N be a TГS-SS ‘N’ of a TГS-S “O”. Afterward kГS-( hГS-) subsemimodule iff . =  Proof: The confirmation involves routine check. A TГS-S ‘O’ is known as cancellative if , , , .k l k m k l m O l m+ = +   = All through the remainder of the paper TГS-S is ‘cancellative’. Definition 3.04: A TГS-S {0} O is called as ‘irreducible’ iff random set ,u v O among u v for any , , , , , ( 1,2,3,4,... & 1,2,3,4,.... , , )i i j j i ik N f g h i S j s i r s r Z  +    = =  such that i i i i j j j j j j j j i i i i i j j i k u f g v h i u h i v f g        + + = +    . A TГS-S ‘O’ is said to be semi- irreducible iff {0}O S S   & O doesn’t have any kГS-subsemimodule except 0 & O. The thoughts of both semi-irreducibility& irreducibility concur by the idea of irreducibility in a TГ- ring S([7],[8],[9]). Proposition 3.05: An ideal Q of a TГ-semiring S & a TГS-S ‘N’ by {0}.N S Q   Then, at that point, the accompanying assertions are valid. (1) If N is semi-irreducible & n N afterward n=0 iff 0 , & ,n s q s S q Q   =      i.e., n=0 iff {0}.N S Q  = (2) If N is irreducible & ,u v N afterward u=v iff 1 1 , , , , , 1,2,.... ; n n i i i i i i i i i i i i i i u k l v k l k l S i p      = = =     =  p is any positive integer. Proof: (1) Given N is semi-irreducible & 0 , & ,n s q s S q Q   =      . Let 0 { : {0}}.N y N y S Q=    = 0.n N  . Let 0.,k l N . Then ( ) {0}.k l S Q k S Q l S Q+      +   = Thus 0.( )k l N+  Thus 0.N is a TГS-subsemimodule of N. Let 0.( ), & .k l l N k N+   Then ( ) 0 & 0 , & , .k l s q l s q q Q s S     + = =      ( ) , & ,k s q k s q l s q k l s q q Q s S          = + = +      hence {0}.k S Q  = Hence 0k N proving 0N is a ternary K-subsemimodule of N. 0{0}, .N S Q N N    S is semi- irreducible subsequently 0 {0}.N =  n=0. On the contrary, if n=0 then 0 , & , .n s q s S q Q   =      (2) Let N be irreducible & ,u v N .u v  {0}, , & 0.N S Q n N t S q Q n t q          Again since N is irreducible, for this n, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 225 https://internationalpubls.com , , , (1 ,1 ; ,i i j jf g h i S i r j s r s Z +       such that 1 1 1 1 . r s s r i i j j j j i i i j j i n u f g v h i u h i v f g        = = = = + + = +    Hence 1 1 1 1 & , , , , , , , , , . r s s r i i j j j j i i i j j i n t p u f g t p v h i t p u h i t p v f g t p t S p P                          = = = = + + = +         1 1 1 1 r s s r i i j j j j i i i j j i n t p u f g v h i u h i v f g          = = = =     + + = +    Where & .i i j jg g t p P i i t p P    =  =  Since N is cancellative and 0n t p   so atleast one of 1 1 1 1 & r r s s i i i i j j j j i i j j u f g v f g u h i v h i        = = = =         holds. The converse part follows easily. Proposition 3.6: Let N be a TГS-S & {0}.N  Subsequently N is semi-irreducible iff for every non- zero ,n N N S S N   = i.e. for any , , , ( 1,2,3,4... & 1,2,3,4.... , , )i i i ik N k l S j s i r s r Z  +    = =  i i i i j j j j i j k n f g n h i    + =  Proof: Let 0N  be semi-irreducible. Then {0}.N S S   Let 0.n N n   Hence by Proposition 3.5, {0};N S S   so {0}.N S S   Since N S S  is a ternary k S -subsemimodule of N, .N S S N   Hence for any {0}.N S S   Hence for any , , , ( 1,2,3,4... & 1,2,3,4.... , , )i i i ik N k l S j s i r s r Z  +    = =  such that i i i i j j j j i j k n f g n h i   + =  In opposition, assume for any ( 0) , .n N n S S N    = Let {0}   is ternary k S -subsemimodule of N. 0.m M n    So, by the given condition .N S S N  = Hence for any for any , , , , , ( 1,2,3,.... & 1,2,3,... , , )i i j j i ik N f g h i S i r j s r s Z  +    = =  such that i i i i j j j j i j k n f g n h i   + =  . Since M is a k S -subsemimodule of N & , , .i i i i j j j j i j n f g n h i M k N       Hence M=N. Now if {0}N S S  = then {0} .n S S n N  =   In particular, {0}N S S  = for any nonzero .n N Hence {0}n S S  = for any ( 0) .n N   N=0- a disagreement. Consequently N is ‘semi-irreducible’. Corollary 3.07: If a TГ-S N is ‘irreducible’, subsequently n S S N  = & semi-irreducible. Proof: Let N be an ITГ-S. {0}N  . ( 0) .n N   any , , , , , ( 1,2,3,... & 1,2,3,.... , , )i i j j i ik N f g h i S j s i r s r Z  +    = =  .i i i i j j j j i j k n f g n h i    + =  N is semi-irreducible TГ-semimodule, by proposition (3.6), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 226 https://internationalpubls.com Subsequently {0} {0}.N S S n S S      = Since n S S  is a ternary k S -subsemimodule of N, .n S S N  = Proposition 3.08: In a TГ-S ‘S’, R exist its ROS. Next N is an ITГS-S iff N is an IR-S. Proof: Let N be an ITГS-S. Characterize R-action on M as go behind: meant for , , [ , ] , , [ , ] .i i i i i i i i i i f g N k R f g k f g k     =   If [ , ] [ , ]i i j j i j k l =  in R then , .i i i j j j i j s t k s t k s t S   =    Since N is an irreducible TГS- semimodule, .n S S N  = (Corollary 3.7). Then for , , , , , , , ( 1,2,... & 1,2,.... ; , )k k t t k k k tn N f g h i N s v S t q k p q p Z  +     = =  .k k k k k t t t t t k t n f g v h i v    + =  So, , , , , , , . (1) . (2) . i i i i k k k k k k i i i t t i i i i k i t i i i i i k k k k k k j j j t t t t t j j j i k j t j j j j j k k k k k k j j j t t t t t j j j j k j t j n k l f g s k l h k l n k l f g s k l h i v k l Again n o p f g s k l h i v k l                           + =  + =  + =          From (1) and (2)N is cancellative. .i i i i j j j j i j n k l n o p   =  Accordingly the R-action define on N is well defined. Currently validate that N as R-semimodule. Let , , .u v N u v  , , , , ,i i j j i i i i i i j j j j j j j j i i i i i j j i k N f g h i S k u f g v h i u h i v f g              + + = +    (using irreducibility of N as TГS-semimodule). [ , , , ] [ , , , ] [ , , , ]i i i i j j j j i i i i i j i k u f g v h i v f g      + + +   where [ , , , ], [ , , , ] .i i i i j j j j i j f g h i R      Consequently N is an IR-S ([6]). On the contrary, presume N is an IR-S. Name Г-action of S on N as go after: for 1 2 1 2, , & , , [ , , , ].f N s s S f S S f s s        =  N is a TГS-S. , , & , [ , , , ], [ , , , ] [ , , , ] [ , , , ] [ , , , ] [ , , , ]. . i i i i j j j j i j i i i i j j j j j j j j i i i i i j j i i i i i j j j j j j j j i i i i i j j i u v N u v k N f g h i R k u f g v h i u h i v f g k u f g v h i u h i v f g                             + + = +  + + = +           S. Consequently by description N is an ITГS-S. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 227 https://internationalpubls.com Let S be a TГ-S. Zeroed of S, is identified as 1 1 1 1 1( ) { : ; }.S y S y z z z S =  + =  evidently, 0 is a associate of Z(S) of a TГ-S ‘S’ among zero (0) component. Z(S) of a TГ-S ‘S’ is an ‘h-ideal’. Permit O is a TГS-S.  (0: O) { : {0}& 0 , }y S o y s o s y o O s S=    =   =     where 1 { : , , , , , } k i i i i i i i i i i i O y s o y s o O y s S k Z    + =   =     . We call (0: O) the annihilator of O in S. Designate it by ( ).SA O A TГS-S ‘O’ is held elect faithful if ( ) ( ).SA O Z S= Proposition 3.09: A TГS-S ‘O’. Afterward ( )SA O is an ‘h-ideal’ of S. Likewise, O is faithful ( / ( ))SS A O - semi module. Proof: Obviously ( )SA O is an additive subsemigroup of S. Currently ( ), , , .Sy A O s S    Next ( ) ( ) {0}.O y S S O y S S    =  = Hence ( )Sy S S A O   verifying as R- ideal. Correspondingly we confirm ( )SA O is a L/La ideal of S. After that 1 2y s z s z+ + = + wherever 1 2, , , ( ).Sy z S s s A O  Afterward 1 2 1 1 2 2, ( ), 0 & 0 & .Sy y A O o t y o y t o t y o y t o O t S         = = = =     Now 1 2 1 2 .y s z s z o t x o t y o t z o t y o t z         + + = +  + + = + This leads to 1 20 0o t y o t y o t y     = = = & O is additively cancellative. Similarly, we can show that 0 & , .o y t o O y t S  =     Thus ( )Sx A O and hence ( )SA O is an ‘h-ideal’ of S. Currently describe a Г-action of / ( )SS A O on O as below: ( / ( )) ( / ( )) ( ) : , , , , / ( ) / ( ).S S S So s A O t A O o S S s t S s A O S A O     =    If / ( ) / ( )S St A O t A O= then 1 1 2 1 1 2 1, ( ) & .St i z t i z i i A O z S+ + = + +    1 2, ( ),Si i A O we have 1 2 0.o s i o s i   = = Now 1 1 2 1 1 1 2 1 , & , , , . t i z t i z o s t o s i o s z o s t o s i o s z o O s S o s t o s t o O                     + + = + +  + + = + +      =    Г-action of / ( )SS A O on O is well defined. Presently seeing that is O is a ( / ( ))SS A O -semimodule. It stays to confirm that / ( ) ( ) ( / ( )). SS A O SA O Z S A O= Clearly / ( )( / ( )) ( ). SS S A OZ S A O A O Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 228 https://internationalpubls.com Now let / ( )/ ( ) ( ) ( / ( )) ( / ( )) 0 & ( / ( )) ( / ( )) 0 . . 0 & 0 & . SS S A O S S S S x A O A O o t A O x A O o x A O t A O i e o t x o x t o O t S        =   = = =    Thus ( )Sx A O and hence Consequently, / ( ) ( / ( ))S Sx A O Z S A O Thus / ( ) ( ) ( / ( )). SS A O SA O Z S A O Hence / ( ) ( ) ( / ( )). SS A O SA O Z S A O Proposition3.10: A TГSS ‘O’& R exist as ROS. Afterward (i) ( )* ( ) & ( )* ( );S R R SA O A O A O A O = = Anywhere M is an ITГS-S (ii) ( ) ( )*, ( )* ( ).S R S R =   =  Proof: ( )* { [ , ] : ( [ , ]) ( )} { [ , ] : ( [ , ]) ( )} { [ , ] : ( [ , ]) {0}} ( ). ( )* { :[ , ] ( )} { : [ , ] {0}} { : {0}} ( ). S i i i i S i i i i i i S i i i i i i R i i R R S A O y R S y A O y R O S T y A O y R M y A O A O y S t A o y S O t y S O T y A O        =   =     =  = = =    =   = =    = =       (ii) Via known 6.14-1proposition & zeroid is an ‘h-ideal’, ( ) ( ( )*)*& ( ) ( ( )*)* .S S R R  =   =  Accordingly demonstrating one of 2 relations is adequate. Permit ( )*.x R Subsequently ( ) [ , ]R x   ( ) ( ).S S x S R S       S have ‘left unity’, ( ).x S ( )* ( ).R S    Now let 1 [ , ] [ , ( )] m i i i x S =    wherever 1, 2,3, 4,.... , ( ).ii m x S =  i i ix z z + = for some .iz S 1 1 1 1 [ , ] [ , ] [ , ] 1,2,3,4.... . [ , ] [ , ] [ , ] [ , ] . i i i i i i m m m i i i i i ii i i m i ii x z z i m x z z where z R        = = = = + =  =  + =      1 [ , ] ( ) m i ii z Z R =  &[ , ( )] ( ).Z S Z R  ( )* ( )Z R Z S  ( ) ( )*.Z S Z R = Proposition3.11: Let S be a TГ-S moreover R exist its ROS. Afterward O is a FTГS-S iff O is a FI R-S. Proof: Let O be a FTГS-S. Subsequently via Proposition 3.8, O is an ITГS-S. Over ( ) ( ).SA O Z S= ( )* ( )* .SA O Z S  = Via 3.10-proposition, ( ) ( ).RA O Z R= Therefore O is a FITГS-S. Speak follows by switching the above contention. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 229 https://internationalpubls.com Definition3.12: A TГ-S S is known to be primitive if it has a FTГS-S. P is an ‘ideal’ of S is entitled Primitive if the Bourne factor TГS ( / )S P is primitive. So a TГS S is ‘primitive’ if {0} is a PI. Lemma3.13: A TГ-S ‘S’ & R is its ROS ‘Q’ be a proper ideal of S. Subsequently ( / ), / *R S Q R Q  are isomorphic, anywhere ( / )R S Q is the ROS of the Bourne factor TГS S/Q. Proposition3.14: Let S be a TГ-S & R be its ROS. If P is a PI of S followed by a PI *P  of R. Proof: Permit P be a PI of S. Afterward S/P is a ‘primitive’ TГS. an irreducible faithfuly ( / )S P -semimodule O. O is a faithful irreducible ( / )R S Q - semimodule via 3.11-proposition wherever ( / )R S Q is the ROS of ( / )S P . / * , ( / )R P R S P are isomorphic(3.13-lemma), O is a FI ( / * )R P  -S. Accordingly, / *R P  is a ‘primitive semiring’ ([6]), ie., *P  is PI. Proposition 3.15 A TГ-S ‘S’ & R exist its ROS. If U is a PI of R afterward *U  is a PI of S. Proof: Presume U is a PI of R. Subsequently R/U is a ‘primitive ternary semiring’. Accordingly,  a FI /R U - S ‘O’. O is a faithful irreducible ( / *)S U -semimodule via 3.11-proposition, So / *S U is a primitive TГS, where U* is a PI of the TГS S. From the over two recommendations & Hypothesis 6.6([1]) the accompanying hypothesis follows without any problem: Theorem3.17: Let S be a TГ-S & R be its ROS. a bijection all PI’s of S & the set of all PI’s of R using the mapping * , →  wherever an ideal P of S. Theorem3.18: A TГ-S ‘S’ is ‘primitive’ iff its ROS ‘R’ is ‘primitive’. Proof: Permit S be a ‘primitive’ TГS.  FTГS-S ‘O’. Subsequently, via 3.11-proposition, O is a FIR-S. Accordingly, R is a primitive semiring -[6]. Contrary follows via reversing the above argument. Ultimately, the accompanying portrayal of ‘primitive h-ideal’ of a TГS closely resembles that of a PI. 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