NON-REDUCED RINGS OF SMALL ORDER AND THEIR MAXIMAL GRAPH

Let r be a commutative ring with nonzero identity. let γ(r) denotes the maximal graph corresponding to the non-unit elements of r, that is, γ(r) is a graph with vertices the non-unit elements of r, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of r containing both. in this paper, we investigate that for a given positive integer n, is there a non-reduced ring r with n non-units? for n ≤ 100, a complete list of non-reduced decomposable rings r = qk i=1 ri (up to cardinalities of constituent local rings ri ’s) with n non-units is given. we also show that for which n, (1 ≤ n ≤ 7500), |center(γ(r))| attains the bounds in the inequality 1 ≤ |center(γ(r))| ≤ n and for which n, (2 ≤ n ≤ 100), |center(γ(r))| attains the value between the bounds.