American International Journal of Research in Science, Technology, Engineering & Mathematics
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ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research)
On Lateral Ternary Γ-Ideals of Ternary Γ-Semirings M. Sajani Lavanya1, Dr. D. Madhusudhana Rao2and V. Syam Julius Rajendra3 1 Lecturer, Department of Mathematics, A.C. College, Gunture, A.P. India. 2 Head, Department of Mathematics, V. S. R & N.V.R. College, Tenali, A.P. India. 3 Lecturer, Department of Mathematics, A.C. College, Guntur, A.P. India. Abstract: In this paper we introduce the terms ternary Γ-semiring, commutative ternary Γ-semiring, left ternary Γ-ideal, lateral ternary Γ-ideal, right ternary Γ-ideal, ternary Γ-ideal, prime lateral ternary Γ-ideal, semiprime ternary Γ-ideal, irreducible ternary Γ-ideal and strongly irreducible ternary Γideal and characterize the ternary Γ-ideals and proved many results. Mathematics Subject Classification: 20G07, 20M10, 20M12, 20M14, 20N10. Key Words: ternary Γ-semiring,lateral ternary Γ-ideal, ternary Γ-ideal, prime lateral ternary Γ-ideal, irreducible.
I. Introduction The literature of ternary algebraic system was introduced by D. H. Lehmer [11] in 1932. He investigated certain ternary algebraic systems called triplexes which turn out to be ternary groups. The notion of ternary semigroups was known to S. Banach. He showed by an example that a ternary semigroup does not necessarily reduce to an ordinary semigroup. In [14], M. L. Santiago developed the theory of ternary semigroups and semiheaps. He devoted his attention mainly to the study of regular ternary semigroups, completely regular ternary semigroups, bi-ideals and intersection ideals in ternary semigroups, the standard embedding of a ternary semigroup and a semiheap with some of their applications. In [12], W. G. Listercharacterized those additive subgroups of rings which are closed under the triple ring product and he called this algebraic system a ternary ring. He also studied the embedding of ternary rings, representation of ternary rings in terms of modules, semisimple ternary rings with minimum condition and radical theory of such rings. D. Madhusudhana Rao, A. Anjaneyulu and A. Gangadhara Rao[1] in 2012 made a study on Γ-ideals in Γ-semigroups. In 2011 [2] they studied about Γradicals in Γ-semigroups. VB Subramanyeswra Rao, A. Anjaneyulu, D. Madhusudhana Rao [15] studied about po-Γ-ideals in po-Γ-semigroups. II. Lateral Ternary �-Ideal Definition II.1: Let T and Γ be two additive commutative semigroups. T is said to be a Ternary �-semiringif there exist a mapping from T ×Γ× T ×Γ× T to T which maps ( x1 ,
ď Ą , x2 , ď ˘ , x3 ) ď‚Ž ď › x1ď Ą x2 ď ˘ x3 ď ? satisfying
the conditions : i) [[ađ?›źbđ?›˝c]Îłdđ?›że] = [ađ?›ź[bđ?›˝cđ?›žd]đ?›że] = [ađ?›źbđ?›˝[cđ?›ždđ?›że]] ii)[(a + b)đ?›źcđ?›˝d] = [ađ?›źcđ?›˝d] + [bđ?›źcđ?›˝d] iii) [a ď Ą (b + c)βd] = [ađ?›źbđ?›˝d] + [ađ?›źcđ?›˝d] iv) [ađ?›źbđ?›˝(c + d)] = [ađ?›źbđ?›˝c] + [ađ?›źbđ?›˝d] for all a, b, c, d∈ T and đ?›ź, đ?›˝, đ?›ž, đ?›żâˆˆ Γ. Obviously, every ternary semiring T is a ternary Γ-semiring. Let T be a ternary semiring and Γ be a commutative ternary semigroup. Define a mapping T ×Γ× T ×Γ× T â&#x;ś T by ađ?›źbđ?›˝c = abc for all a, b, c ∈ T and đ?›ź, đ?›˝ ∈ Γ. Then T is a ternary Γ-semiring. Definition II.2: An element 0of a ternary Γ-semiring T is said to be an absorbingzero of T provided 0 + x = x = x + 0and 0đ?›źađ?›˝b = ađ?›ź0βb = ađ?›źbđ?›˝0 = 0  a, b, x ďƒŽ T and đ?›ź, đ?›˝âˆˆÎ“. DefinitionII.3 : Let T be ternary Γ-semiring. A non empty subset ‘S’ is said to be a ternary subđ?šŞ-semiring of T if S is an additive subsemigroup of T and ađ?›źbđ?›˝c ďƒŽ S for all a,b,c ďƒŽ S and đ?›ź, đ?›˝âˆˆÎ“. NoteII.4 : A non empty subset S of a ternary Γ-semiring T is a ternary subΓ-semiring if and only if S + S ⊆ S and SΓSΓS ďƒ? S. DefinitionII.5 : A nonempty subset A of a ternary Γ-semiring T is said to be left ternary đ?šŞ-ideal of T if (1) a, b ∈ A implies a + b ∈ A. (2) b, c ďƒŽ T, a ďƒŽ A, đ?›ź, đ?›˝âˆˆÎ“ implies bđ?›źcđ?›˝a ďƒŽ A.
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