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Non-reduced rings of small order and their maximal graph | ||
| Journal of Algebra and Related Topics | ||
| مقاله 3، دوره 6، شماره 1، شهریور 2018، صفحه 35-44 اصل مقاله (320.11 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2018.10130.1097 | ||
| نویسندگان | ||
| A. Sharma* ؛ A. Gaur | ||
| Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India | ||
| چکیده | ||
| 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 | ||
| کلیدواژه‌ها | ||
| Non-reduced ring؛ Jacobson radical؛ maximal graphs؛ center؛ median | ||
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آمار تعداد مشاهده مقاله: 721 تعداد دریافت فایل اصل مقاله: 949 |
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