American J. of Mathematics and Sciences Vol. 3, No -1 ,(January 2014) Copyright  Mind Reader Publications ISSN No: 2250-3102 131 ON CENTER OF FINITELY GENERATED LOCALLY (-1,1) RINGS K.Jayalakshmi Assistant Professor in Mathematics, J.N.T.University Anantapur College of Engg. J.N.T.University Anantapur. Anantapur.(A.P) INDIA. jayalakshmikaramsi@gmail.com C.Manjula Department of Mathematics, J.N.T.University Anantapur College of Engg. J.N.T.University Anantapur. Anantapur.(A.P) INDIA. man7ju@gmail.com ABSTRACT : A simple finitely generated locally (-1,1) ring must be an associative field. 2010 MATHEMATICS SUBJECT CLASSIFICATION : 17D20 KEY WORDS : Locally (-1,1) ring, nilpotent ideal, simple ring. INTRODUCTION : Hentzel and smith [1] studied simple locally (-1,1) nil rings and showed that a simple locally (-1,1) nil ring of char. ≠ 2,3 must be associative. Hentzel [1] studied properties of nil potent ideals in semi simple (- 1,1) rings which are nil. We concentrate mainly on the result of Hentzel [1] and prove that a simple finitely generated locally (-1,1) ring must be an associative field. A ring is a (-1,1) ring if it is satisfies the conditions: 0 ≡ A(x,y,z) = (x,y,z) + (y,z,x) + (z,x,y). … (1) 0 ≡ B(x,y,z) = (x,y,z) + (x,z,y). … (2) A ring is locally (-1,1) if the subring generated by any two of its elements is (-1,1). For example, both (-1,1) rings and alternative rings are locally (-1,1). In a nonassociative ring R, we define (x,y,z) = (xy)z – x(yz) and [x,y] = xy – yx for all x,y ∈ R. A ring R is said to be simple if whenever A is an ideal of R then either A = R or A = 0. By the center Z of R we mean that the set of all elements z in N such that [z,R] = 0 i.e., Z = {z ∈ N / [z,R] = 0}. That is C represents set of all elements which commutes with all elements in the ring and c will always means and elements taken from C. We use the following identities which hold in locally (-1,1) ring char. ≠ 2,3, which were proved by Hentzel [1]. Whereas the commutative center C is defined as C = {c ∈ R / [c,R] = 0}. 0 ≡ C(x,y,z) = (x,y,yz) – (x,y,z)y. … (3) 0 ≡ D(x,y,z,w) = (x,yz,w) + (x,wz,y) – (x,z,w)y – (x,z,y)w. … (4)