| Title:
|
Some results on the annihilator graph of a commutative ring (English) |
| Author:
|
Afkhami, Mojgan |
| Author:
|
Khashyarmanesh, Kazem |
| Author:
|
Rajabi, Zohreh |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
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67 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
151-169 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace $. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \geq 1$. (English) |
| Keyword:
|
annihilator graph |
| Keyword:
|
zero-divisor graph |
| Keyword:
|
outerplanar |
| Keyword:
|
ring-graph |
| Keyword:
|
cut-vertex |
| Keyword:
|
clique number |
| Keyword:
|
weakly perfect |
| Keyword:
|
chromatic number |
| Keyword:
|
polynomial ring |
| Keyword:
|
ring of fractions |
| MSC:
|
05C75 |
| MSC:
|
05C99 |
| MSC:
|
13A99 |
| idZBL:
|
Zbl 06738510 |
| idMR:
|
MR3633004 |
| DOI:
|
10.21136/CMJ.2017.0436-15 |
| . |
| Date available:
|
2017-03-13T12:08:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146046 |
| . |
| Reference:
|
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