Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE STRUCTURE OF ZERO-DIVISOR ELEMENTS IN A NEAR-RING OF SKEW FORMAL POWER SERIES

  • Alhevaz, Abdollah (Faculty of Mathematical Sciences Shahrood University of Technology) ;
  • Hashemi, Ebrahim (Faculty of Mathematical Sciences Shahrood University of Technology) ;
  • Shokuhifar, Fatemeh (Faculty of Mathematical Sciences Shahrood University of Technology)
  • Received : 2019.12.15
  • Accepted : 2020.11.03
  • Published : 2021.04.30
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Abstract

The main purpose of this paper is to study the zero-divisor properties of the zero-symmetric near-ring of skew formal power series R0[[x; α]], where R is a symmetric, α-compatible and right Noetherian ring. It is shown that if R is reduced, then the set of all zero-divisor elements of R0[[x; α]] forms an ideal of R0[[x; α]] if and only if Z(R) is an ideal of R. Also, if R is a non-reduced ring and annR(a - b) ∩ Nil(R) ≠ 0 for each a, b ∈ Z(R), then Z(R0[[x; α]]) is an ideal of R0[[x; α]]. Moreover, if R is a non-reduced right Noetherian ring and Z(R0[[x; α]]) forms an ideal, then annR(a - b) ∩ Nil(R) ≠ 0 for each a, b ∈ Z(R). Also, it is proved that the only possible diameters of the zero-divisor graph of R0[[x; α]] is 2 and 3.

Keywords

Acknowledgement

This research was in part supported by a grant from Shahrood University of Technology.

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