Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON SOME TYPE ELEMENTS OF ZERO-SYMMETRIC NEAR-RING OF POLYNOMIALS

  • Hashemi, Ebrahim (Faculty of Mathematical Sciences Shahrood University of Technology) ;
  • Shokuhifar, Fatemeh (Faculty of Mathematical Sciences Shahrood University of Technology)
  • Received : 2018.02.24
  • Accepted : 2018.07.16
  • Published : 2019.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative ring with unity. In this paper, we characterize the unit elements, the regular elements, the ${\pi}$-regular elements and the clean elements of zero-symmetric near-ring of polynomials $R_0[x]$, when $nil(R)^2=0$. Moreover, it is shown that the set of ${\pi}$-regular elements of $R_0[x]$ forms a semigroup. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its "multiplication" operation.

Keywords

References

  1. D. F. Anderson and A. Badawi, Von Neumann regular and related elements in commutative rings, Algebra Colloq. 19 (2012), Special Issue no. 1, 1017-1040. https://doi.org/10.1142/S1005386712000831
  2. J. C. Beidleman, A note on regular near-rings, J. Indian Math. Soc. (N.S.) 33 (1969), 207-209 (1970).
  3. G. F. Birkenmeier and F. Huang, Annihilator conditions on polynomials, Comm. Algebra 29 (2001), no. 5, 2097-2112. https://doi.org/10.1081/AGB-100002172
  4. D. Z. T. Chao, A radical of unitary near-rings, Tamkang J. Math. 6 (1975), no. 2, 293-299.
  5. Y. U. Cho, Some results on monogenic (R, S)-groups, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 18 (2011), no. 4, 305-312.
  6. E. Hashemi, On nilpotent elements in a nearring of polynomials, Math. Commun. 17 (2012), no. 1, 257-264.
  7. E. Hashemi and A. A. Estaji, On polynomials over abelian rings, Vietnam J. Math. 40 (2012), no. 1, 47-55.
  8. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.
  9. C. J. Maxson, On local near-rings, Math. Z. 106 (1968), 197-205. https://doi.org/10.1007/BF01110133
  10. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
  11. G. Pilz, Near-Rings, second edition, North-Holland Mathematics Studies, 23, North-Holland Publishing Co., Amsterdam, 1983.