| Title:
|
Classification of rings with toroidal Jacobson graph (English) |
| Author:
|
Selvakumar, Krishnan |
| Author:
|
Subajini, Manoharan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
307-316 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative ring with nonzero identity and $J(R)$ the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by $\mathfrak J_R$, is defined as the graph with vertex set $R\setminus J(R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1-xy$ is not a unit of $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_n$. In this paper, we investigate the genus number of the compact Riemann surface in which $\mathfrak J_R$ can be embedded and explicitly determine all finite commutative rings $R$ (up to isomorphism) such that $\mathfrak J_R$ is toroidal. (English) |
| Keyword:
|
planar graph |
| Keyword:
|
genus of a graph |
| Keyword:
|
local ring |
| Keyword:
|
nilpotent element |
| Keyword:
|
Jacobson graph |
| MSC:
|
05C10 |
| MSC:
|
05C25 |
| MSC:
|
13M05 |
| idZBL:
|
Zbl 06604468 |
| idMR:
|
MR3519603 |
| DOI:
|
10.1007/s10587-016-0257-y |
| . |
| Date available:
|
2016-06-16T12:37:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145725 |
| . |
| Reference:
|
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