An Alternative Perspective of Near-rings of Polynomials and Power series.

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if annS(a) = annS(b), is an equivalence relation. The compressed zero-divisor graph ΓE(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]S and [1]S, in which two distinct vertices [a]S and [b]S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R0[x] and the near-ring of the power series R0[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(ΓE(R0[x])) and diam(ΓE(R0[[x]])). As a corollary, it is shown that for a reduced ring R, diam(ΓE(R)) ≤ diam(ΓE(R0[x])) ≤ diam(ΓE(R0[[x]])). [ABSTRACT FROM AUTHOR]