www.tjprc.org editor@tjprc.org International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 4, Issue 3, Jun 2014, 17-32 © TJPRC Pvt. Ltd. A STUDY ON THE STRUCTURE OF RIGHT TERNARY N-GROUPS A. UMA MAHESWARI 1 & C. MEERA 2 1 Department of Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai, Tamil Nadu, India 2 Department of Mathematics, Bharathi Women’s College (Autonomous), Chennai, Tamil Nadu, India ABSTRACT Right ternary near-ring (RTNR) is a generalisation of its binary counterpart. In this paper right ternary N-group (or N-module) of an RTNR ‘N’ is defined and its basic algebraic properties are given. The substructures of a right ternary N-group (N-subgroups, normal subgroups, ideals) and the factor N-groups are also defined and homomorphism theorems on right ternary N-groups are obtained. The concept of faithful right ternary N-groups and monogenic right ternary N-groups are given in this generalised setting and every commutative RTNR is realized as M(Γ) (an RTNR of all mappings of an additive group Γ) where Γ is a faithful right ternary N-group. A simple monogenic right ternary N-group Γ is characterised in terms of maximal left ideals of N. If Γ is a faithful IFP right ternary N-group then N is shown as an IFP-RTNR. The three types of N-groups are defined and the relationships among them are established. AMS Classification: 20N10, 16Y30, 16D10, 16D25 KEYWORDS: Zero-Symmetric RTNR, Constant RTNR, Biunital Element, Normal Subgroup 1. INTRODUCTION Near-rings are appropriate structures to study non-linear functions on finite groups. The set of all functions on groups under pointwise addition and composition are typical examples of near-rings. Just in the same way as R-modules over a ring R are used in ring theory, N-groups play an important role in the theory of near-rings. The fundamental properties of algebraic structures can deeply be understood and further be developed in their n-ary context. Ternary algebraic structures [1, 3] have applications in Mathematical and theoretical physics. Lister [1] characterized additive subgroups of rings which are closed under triple ring product. The authors introduced right ternary near-rings (RTNR) [7] and have studied their properties [4]. In this paper N-groups of an RTNR N are defined and their algebraic properties are studied. The substructures of N-group namely N-subgroups and ideals are considered. The factor right ternary N-groups are defined and homomorphism theorems on right ternary N-groups are obtained. It is proved that in a zero-symmetric RTNR every ideal of a right ternary N-group is an N-subgroup. The kernel of an N- homomorphism and the image of an onto N-homomorphism are shown as ideals. The definition of Noetherian quotient of two subsets of a right ternary N-group and the basic properties as given in [2] are established in this generalized setting. The faithful right ternary N-groups and monogenic right ternary N-groups are defined and the process of embedding a commutative RTNR in M(Γ) (an RTNR of all mappings of Γ) where Γ is a faithful right ternary N-group is