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http://dx.doi.org/10.4134/CKMS.c190433

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)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.2, 2021 , pp. 197-207 More about this Journal
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
Symmetric ring; ${\alpha}$-compatible ring; near-ring of skew formal power series; zero-divisor element;
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Times Cited By KSCI : 1  (Citation Analysis)
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