**InternationaleJournals**

**Primary Ideals in Ternary Semigroups**

^{1}**G. Hanumanta Rao, ^{2}A. Anjaneyulu and ^{3}D. Madhusudhana Rao**

^{1}Department of Mathematics, S.V.R.M. College, Nagaram, Guntur (dt) A.P. India. Email : ghr@svrmc.edu.in

^{2, 3}Department of Mathematics, V.S.R & N.V.R.College, Tenali, A.P. India. Email: anjaneyulu.addala@gmail.com, raoag1967@gmail.com

**ABSTRACT**

In this paper, the terms left primary ideal, lateral primary ideal, right primary ideal, primary ideal, left primary ternary semigroup, lateral primary ternary semigroup , right primary ternary semigroup, primary ternary semigroup are introduced. It is proved that A be an ideal in a ternary semigroup T and if X, Y, Z are three ideals of T such that 1) XYZA and Y ⊈ A, Z ⊈ A, implies X A A iff x, y, z T, < x > < y > < z > A and y A, z A implies x A . 2) XYZ A and X ⊈A, Z ⊈A, implies Y A if and only if x, y, z T, < x >< y >< z > A and xA, zA implies y A . 3) XYZ A and X ⊈ A, Y ⊈ A, implies Z A iff x, y, z T, < x >< y >< z > A and xA, yA implies z A . Further it is proved that if T be a commutative ternary semigroup and A be an ideal of T, then the conditions, 1) A is left primary ideal, 2) X, Y, Z are three ideals of T such that XYZA and Y ⊈ A, Z ⊈ A, implies X A , 3) x, y, z T, < x >< y >< z > A and y A, z A implies x A are equivalent. It is also proved that if T be a commutative ternary semigroup and A be an ideal of T, then the conditions, 1) A is lateral primary ideal, 2) X, Y, Z are three ideals of T such that XYZ A and X ⊈ A, Z ⊈A, implies Y A , 3) x, y, z T, < x >< y >< z > A and x A, z A implies y A . Further the conditions for an ideal in a commutative ternary semigroup T, 1) A is right primary ideal, 2) X, Y, Z are three ideals of T such that XYZ A and X ⊈A, Y ⊈A, implies Z A , 3) x, y, z T, x y z A and x A, y A implies z A are equivalent. It is proved that every ideal A in a ternary semigroup T, 1) T is a left primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ A and Y ⊈A, Z ⊈ A, implies X A , 2) T is a lateral primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ A and X ⊈A, Z ⊈A, implies Y A , 3) T is a right primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ A and X ⊈A, Y ⊈A, implies Z A . It is proved that T be a ternary semigroup with identity and M be the unique maximal ideal in T. If A = M for some ideal A in T, then A is a primary ideal. Further it proved that if T is a ternary semigroup with identity and M is the unique maximal ideal of T , then for any odd natural number n, M n is a primary ideal of T. It is proved that if A is an ideal of quasi commutative ternary semigroup T, then 1) A is primary, 2) A is left primary, 3) A is lateral primary and 4) A is right primary are equivalent.

**Subject Classification(2010):** 20M07, 20M11, 20M12.

**Key Words:** Left primary ideal, Lateral primary ideal, right primary ideal, primary ideal, left primary ternary semigroup, lateral primary ternary semigroup , right primary ternary semigroup, primary ternary semigroup.

**International
eJournal of Mathematics and Engineering**

**Volume 4,
Issue ****3,** **Pages: ** 2145 - 2159