(–1, 1) Rings without Nilpotent Elements

  • Authors

    • K. Hari babu
    • K. Jayalakshmi
    2018-10-02
    https://doi.org/10.14419/ijet.v7i4.10.20943
  • Nonassociative ring, (–1, 1) ring, simple ring, idempotent, nilpotent.

  • Abstract

    A ring of  type 12(خ³,خ´)"> , was introduced by Albert and Kokoris [1,3] where in they have shown that a simple ring of 12(خ³,خ´)">  type is either associative or contains no idempotent other than 1. In this paper we obtain further results on the residual cases, to prove that a nonassociative (-1,1) rings satisfying (x, x, y)2 = 0, for all elements of the rings imply (x, x, y) = 0. But then indeed (-1,1) rings which have no nilpotent elements are associative and there by all such rings are division rings.

     

     

  • References

    1. [1] A. A. Albert, Almost alternative algebras, Portugaliae math. 8 (1949), pp.23-36.

      [2] E. Kleinfeld, Right alternative rings, Proc. Amer. Math .Soc.4 (1953), pp.939-944.

      [3] L.A.Kokoris, On a class of almost alternative algebras, canad. J. of. Math. 8 (1956), pp.250- 255.

      [4] I. R. Hentzel, The Characterization of (–1, 1) rings, J. Algebra (1972) pp.236- 258.

      [5] K. Suvarna, and C. Jaya Subba Reddy., A Generalization of (1, 1) Rings, Acta Ciencia Indica. (2007), 429.

      [6] K. Jayalakshmi and K. Hari Babu., On (-1, 1) rings Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 945-954.

      [7] Yu.A.Bakhturin, I.P.Shestako and M. Slin'ko NONASSOCIATIVE RINGS Volume 18, Issue 2, January 22, 1982 page no: 169-211.

      [8] L. Sreenivasulu Reddy, T. Mahesh Kumar, and C. Jaya Subba Reddy The Fundamental Results on Non-Associative Rings with Cyclic Property International Journal of Engineering and Advanced Technology (IJEAT) June 2013, Vol. 2

      [9] K Madhusudhan Reddy, Commutativity of nonassociative rings with identities in the Center, IOP Conf. Series (2017), Materials Science and Engineering 263.

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  • How to Cite

    Hari babu, K., & Jayalakshmi, K. (2018). (–1, 1) Rings without Nilpotent Elements. International Journal of Engineering & Technology, 7(4.10), 387-388. https://doi.org/10.14419/ijet.v7i4.10.20943

    Received date: 2018-10-04

    Accepted date: 2018-10-04

    Published date: 2018-10-02