ON ZERO-DIVISORS OF NEAR-RINGS OF POLYNOMIALS

摘要

In this paper, we are interested to study zero-divisor properties of a 0-symmetric nearring of polynomials R-0[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R-0[x], namely Z(R-0[x]), is an ideal of R-0[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(R-0[x]) is an ideal of Z(R-0[x]) if and only if ann(R)({a, b}) boolean AND N il(R) not equal 0, for each a, b is an element of Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R-0[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R-0[x] is the graph Gamma(R-0[x]) such that the vertices of Gamma(R-0[x]) are all the non-zero zero-divisors of R-0[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 Gamma(R-0[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.