Non-reduced rings of small order and their maximal graph

Let R be a commutative ring with nonzero identity. Let \Gamma(R) denotes the maximal graph corresponding to the non-unit elements of R, that is, \Gamma(R)is a graph with vertices the non-unit elements of R, where two distinctvertices a and b are adjacent if and only if there is a maximal ideal of Rcontaining 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 \leq 100, a complete list of non-reduced decomposable rings R = \prod_{i=1}^{k}R_i (up to cardinalities of constituent local rings R_i's) with n non-units is given. We also show that for which n, (1\leq n \leq 7500), |Center(\Gamma(R))| attains the bounds in the inequality 1\leq |Center(\Gamma(R))|\leq n and for which n, (2\leq n\leq 100), |Center(\Gamma(R))| attains the value between the bounds

Non-reduced rings of small order and their maximal graph Keywords:

Non-reduced ring , Jacobson radical , maximal graphs , center , median