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On structure and commutativity of certain periodic near rings

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Abstract

In the present paper we first establish decomposition theorems for near rings satisfying either of the properties xy = xmypxn or xy = ymxpyn, where m≥1, n≥1, p≥1 are positive integers depending on the pair of near ring elements x,y; and further, we investigate commutativity of such near rings. Moreover, it is also shown that under some additional hypotheses, such nearrings turn out to be commutative rings.

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References

  1. Asma Ali, M. Ashraf and M.A. Quadri, Some elementary commutativity conditions for near rings, Math. Student 56(1988), 181–183

    MathSciNet  MATH  Google Scholar 

  2. H.E. Bell, Near rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2(1970), 363–368

    MathSciNet  MATH  Google Scholar 

  3. H.E. Bell, Certain near rings are rings, J. London Math. Soc. 4(1971), 264–270

    Article  MathSciNet  MATH  Google Scholar 

  4. H.E. Bell, Centres for near rings: Applications to commutativity theorems, Proc. Edinburgh Math. Soc. 23(1971), 264–270

    MathSciNet  Google Scholar 

  5. H.E. Bell and S. Ligh, Some decomposition theorems for periodic rings and near rings, Math. J. Okayama Univ. 31(1989), 93–99

    MathSciNet  MATH  Google Scholar 

  6. J.R Clay, The near rings on groups of low order, Math. Z. 104(1968), 364–371

    Article  MathSciNet  MATH  Google Scholar 

  7. C.G. Fain, Some structure theorems for near rings, (Doctoral dissertation, University of Oklahoma 1986).

  8. A. Frohlich, Distributively generated near rings, Proc. London Math. Soc. 8(1958), 76–94

    Article  MathSciNet  MATH  Google Scholar 

  9. I.N. Herstein, A note on rings with central nilpotent elements, Proc. Amer. Math. Soc. 5(1954), 620

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Jacobson, Structure theory of algebraic algebras of bounded degreee, Ann. of Math. 46(1945), 695–707

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Ligh and J. Luh, Some commutativity theorems for rings and near rings, Acta Math. Acad. Sci. Hungar. 28(1976), 19–23

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Ligh and J. Luh, Direct sum of J-rings and zero rings, Amer. Math. Monthly 96 (1989), 40–41

    Article  MathSciNet  MATH  Google Scholar 

  13. M.A. Quadri, M. Ashraf and Asma Ali, Certain conditions under which near rings are rings, Bull. Austral. Math. Soc. 42(1990), 91–94

    Article  MathSciNet  MATH  Google Scholar 

  14. M.A. Quadri, M. Ashraf and Asma Ali, Certain conditions under which near rings are rings-II, Rad. Mat. (to appear).

  15. M.O. Searcoid and D. MacHale, Two elementary generalizations for Boolean rings, Amer. Math Monthly 93(1986), 121–122

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Taussky, Rings with non-commutative addition, Bull. Calcutta Math. Soc. 28(1936), 245–246

    Google Scholar 

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  1. Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

    Mohammad Ashraf

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  1. Mohammad Ashraf
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Ashraf, M. On structure and commutativity of certain periodic near rings. Results. Math. 24, 201–210 (1993). https://doi.org/10.1007/BF03322330

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