| Title:
|
The prime ideals intersection graph of a ring (English) |
| Author:
|
Nikmehr, M. J. |
| Author:
|
Soleymanzadeh, B. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
137-145 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative ring with unity and $U(R)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of $R$, denoted by $G_{p}(R)$. The $G_{p}(R)$ is a graph with vertex set $R^*-U(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $\mathfrak{p}$ of $R$ such that $a,b\in \mathfrak{p}$. We obtain necessary and sufficient conditions on $R$ such that $G_{p}(R)$ is disconnected. We find the diameter and girth of $G_{p}(R)$. We also determine all rings whose prime ideals intersection graph is a star, path, or cycle. At the end of this paper, we study the planarity and outerplanarity of $G_{p}(R)$. (English) |
| Keyword:
|
the prime ideals intersection graph of a ring |
| Keyword:
|
clique number |
| Keyword:
|
planar graph |
| MSC:
|
05C40 |
| MSC:
|
05C69 |
| MSC:
|
13A15 |
| idZBL:
|
Zbl 06773709 |
| idMR:
|
MR3666936 |
| DOI:
|
10.14712/1213-7243.2015.205 |
| . |
| Date available:
|
2017-06-13T13:21:31Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146782 |
| . |
| Reference:
|
[1] Akbari S., Nikandish R., Nikmehr M.J.: Some results on the intersection graphs of ideals of rings.J. Algebra Appl. 12 (2013), no. 4, 1250200. Zbl 1264.05056, MR 3037274, 10.1142/S0219498812502003 |
| Reference:
|
[2] Anderson D.F., Badawi A.: The generalized total graph of a commutative ring.J. Algebra Appl. 12 (2013), no. 5, 1250212. Zbl 1272.13003, MR 3055569, 10.1142/S021949881250212X |
| Reference:
|
[3] Anderson D.F., Badawi A.: On the total graph of a commutative ring without the zero element.J. Algebra Appl. 11 (2012), no. 4, 1250074. Zbl 1247.13004, MR 2959423, 10.1142/S0219498812500740 |
| Reference:
|
[4] Anderson D.F., Badawi A.: The total graph of a commutative ring.J. Algebra 320 (2008), no. 7, 2706–2719. Zbl 1247.13004, MR 2441996, 10.1016/j.jalgebra.2008.06.028 |
| Reference:
|
[5] Anderson D.F., Livingston P.: The zero-divisor graph of a commutative ring.J. Algebra 217 (1999), no. 2, 434–447. Zbl 1035.13004, MR 1700509, 10.1006/jabr.1998.7840 |
| Reference:
|
[6] Atiyah M.F., Macdonald I.G.: Introduction to Commutative Algebra.Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1969. Zbl 0238.13001, MR 0242802 |
| Reference:
|
[7] Beck I.: Coloring of commutative rings.J. Algebra 116 (1988), no. 1, 208–226. Zbl 0654.13001, MR 0944156, 10.1016/0021-8693(88)90202-5 |
| Reference:
|
[8] Corbas B., Williams G.D.: Rings of order $p^5$, I. Nonlocal rings.J. Algebra 231 (2000), 677-690. Zbl 1017.16014, MR 1778165, 10.1006/jabr.2000.8349 |
| Reference:
|
[9] Raghavendran R.: Finite associative rings.Compositio Math. 21 (1969), 195–229. Zbl 0179.33602, MR 0246905 |
| Reference:
|
[10] Syslo M.M.: Characterizations of outerplanar graphs.Discrete Math. 26 (1979), no. 1, 47–53. Zbl 0401.05040, MR 0535083, 10.1016/0012-365X(79)90060-8 |
| Reference:
|
[11] West D.B.: Introduction to Graph Theory.second edition, Prentice Hall, Upper Saddle River, 2001. Zbl 1121.05304, MR 1367739 |
| Reference:
|
[12] Lu D., Wu T.: On bipartite zero-divisor graphs.Discrete Math. 309 (2009), no. 4, 755–762. MR 2502185, 10.1016/j.disc.2008.01.044 |
| . |
