On zero-divisors of near-rings of polynomials

Abstract

In this paper, we are interested to study zero-divisor properties of a 0- symmetric nearring of polynomials R₀[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R₀[x], namely Z(R₀[x]), is an ideal of R₀[x] if and only if Z(R) is an ideal of R and R has Property (A). For a non-reduced ring R, it is shown that Z(R₀[x]) is an ideal of Z(R₀[x]) if and only if ann_R({a; b}) ∩ Niℓ(R) ≠ 0, for each a; b ∈ Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R₀[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R₀[x] is the graph Γ(R₀[x]) such that the vertices of Γ(R₀[x]) are all the non-zero zero-divisors of R₀[x] and two distinct vertices f and g are connected by an edge if and only if f ◦ g = 0 or g ◦ f = 0. Among other results, we give a complete characterization of the possible diameters of Γ(R₀[x]) in terms of the ideals of R. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its "multiplication" operation.

Mathematics Subject Classification (2010): Primary: 16Y30; Secondary: 05C12.

Keywords: Near-ring of polynomials, zero-divisors, graph, diameter, girth