Journal of Science and Arts Year 11, No. 4(17), pp. 367-368, 2011 ORIGINAL PAPER COMMUTATIVITY OF SEMI NEAR RINGS VOLETY V.S. RAMACHANDRAM 1 _________________________________________________ Manuscript received: 05.08.2011; Accepted paper: 12.10.2011; Published online: 01.12.2011 Abstract. In this paper we have mainly obtained some results related to commutativity of cancellative semi near rings with identity element. Keywords: Semi near ring, Cancellative semi near ring, identity element. AMS subject classification (2000): 16Y60, 16W50. 1. INTRODUCTION V. G. Van Hoorn and B. Van Rootselaar [4] discussed general theory of semi near rings. In this paper we provided some necessary and sufficient conditions for the commutativity of cancellative semi near rings with identity element. 2. PRELIMINARIES Definition 1: A nonempty set N together with two binary operations ‘+’ and ‘.’ is said to be a semi near ring if (N, +) is a semi group and (N, .) is a semi group satisfying the distributive laws. Definition 2: A semi near ring N is cancellative if the following conditions hold . ,, abc N   (1) a b a c b c      (2) b a c a b c      Definition 3: In a semi near ring N if there exists an element ‘e’ such that then N is called a semi near ring with identity element. . . ae ea a   a N  We start with the following theorem. Theorem 1: A cancellative semi near ring N with identity is commutative if and only if 2 . .. x y yx  y , x y N   . . Proof: First part is trivial. For the converse replace theelement y with y + e. We get , 2 .( ) ( ). .( ) x y e y exy e     .( ).( ) (. )( ) x y e y e yx x y e       ( ).( ) ( . ). . . xy x y e yx y yx xy x        ( . ). . . ( . ). . . x y y xy xy x yx y yx xy x         ( . ). . ( . ). . x y y xy yx y yx     (By Cancellation law) (.) . .. . x yy xy yxy yx     ( By Associativity) 2 . . .. . x y xy yxy y     x . (By the given condition) . x y yx   , x y N   Hence N is a commutative semi near ring. 1 Bonam Venkata Chalamayya (B.V.C.) College of Engineering, Palacharla, Rajahmundry, 533104, E.G. District, AndhraPradesh, India. E-mail: vvsrch@yahoo.com . ISSN: 1844 – 9581 Mathematics Section