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.