Non-reduced rings of small order and their maximal graph

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India


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 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 $n \leq 100$, 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