@article {
author = {Sharma, A. and Gaur, A.},
title = {Non-reduced rings of small order and their maximal graph},
journal = {Journal of Algebra and Related Topics},
volume = {6},
number = {1},
pages = {35-44},
year = {2018},
publisher = {University of Guilan},
issn = {2345-3931},
eissn = {2382-9877},
doi = {10.22124/jart.2018.10130.1097},
abstract = {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 $R$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},
keywords = {Non-reduced ring,Jacobson radical,maximal graphs,center,median},
url = {https://jart.guilan.ac.ir/article_3025.html},
eprint = {https://jart.guilan.ac.ir/article_3025_8bced734856b35561fe82af4e15d0d5c.pdf}
}