ALTERNATOR IDEALS IN VINBERG (-1, 1) RINGS K. JAYALAKSHMI 1 AND K. HARI BABU 2 Abstract. In this paper we characterize a 2, 3-torsion free nonassociative Vinberg (-1, 1) ring R satisfying third power associative conditions. In such ring the square of an alternator ideal is trivial, if the ring is nil of index n that does not have elements of order atmost n then the ring is solvable of index atmost n(n+1) 2 + 1 and is locally nilpotent. AMS Subject Classification: 17D20, 17A30 and 17A36. Key Words and Phrases:Vinberg ring, (-1, 1) ring, al- ternator ideal, solvable, locally nilpotent. 1. Introduction Outcalt, Stirling and Elisabeth [6,8 and 2] studied the properties of Vinberg rings. Pchelintsev [7] established some properties of alternator ideals of finitely generated binary (-1, 1) algebras to show that the algebras are solvable. For more details on Vinberg rings and the properties of nilpotency of (-1, 1) rings see [4, 5 and 8]. Using the concepts of above authors we characterize Vinberg (-1, 1) ring by a central extension process. The associators of the form (x, x, y)and (x, y, x) are equal in Vinberg (-1, 1) rings. M and E denotes an alternator subring and an alternator ideal, respectively. We prove that E = M + MR and E 2 = 0. Throughout this paper R denotes a 2,3-torsion free Vinberg (-1, 1) ring unless specified. If R satisfies third power associativity then it is alternative. Since the quotient ring R/E is alternative, we follow Zhevlakov’s theorem (see [9, p. 158]) on solvability and Shirshov’s theorem (see [9, pp. 148, 150]) on local nilpotency of alternative nilrings of bounded index and generalize it to the class of 2,3-torsion free Vinberg (-1, 1) ring. 1991 Mathematics Subject Classification. 17D20, 17A30 and 17A36. Key words and phrases. Vinberg ring, (-1, 1) ring, alternator ideal, solvable, locally nilpotent. 1 International Journal of Pure and Applied Mathematics Volume 113 No. 6 2017 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu ijpam.eu 2017