Yıl 2018, Cilt 23, Sayı 23, Sayfalar 157 - 166 2018-01-11
| | | |

## Some graph on prime ideals of a commutative ring

#### Alpesh M. Dhorajia [1]

##### 70 93

Let R be a commutative ring with an identity. Let Spec(R) be
the set of all prime ideals of R and Max(R) be the set of all maximal ideals
of R. Let S  Max(R). We de ne an S-proper ideal sum graph on Spec(R),
denoted by 􀀀S(Spec(R); S), as an undirected graph whose vertex set is the set
Spec(R) and, for two distinct vertices P and Q, there is an arc from P to Q,
whenever P +Q M, for some maximal idealMin S. In this paper, we prove
that the complement graph of a proper sum graph 􀀀(Spec(R); S) is complete
if and only if R is an Artinian ring. We also study some basic properties of
the graph 􀀀S(Spec(R); S) such as connectivity, girth and clique number. We
explore the in
uence of the ring theoretic properties of a commutative ring R
on the proper sum graph of R and vice versa.
Proper sum graph, Artinian ring, commutative ring
• S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra, 274(2) (2004), 847-855.
• D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36(8) (2008), 3073-3092.
• D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7) (2008), 2706-2719.
• D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
• M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Alge- bra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
• I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226. [7] N. Bloom eld and C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra, 38(8) (2010), 2965-2980.
• J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976 I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309(17) (2009), 5381-5392.
• A. M. Dhorajia, Total graph of the ring Zn  Zm, Discrete Math. Algorithms Appl., 7(1) (2015), 1550004 (9 pp).
• A. M. Dhorajia and J. M. Morzaria, Domination in total graphs of small rings, Discrete Math. Algorithms Appl., 8(4) (2016), 1650069 (11 pp).
• H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34(3) (2006), 923-929.
• H. R. Maimani, C. Wickham and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math., 42(5) (2012), 1551-1560.
• M. J. Nikmehr and F. Shaveisi, The regular digraph of ideals of a commutative ring, Acta Math. Hungar., 134(4) (2012), 516-528.
• P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176(1) (1995), 124-127.
• S. Spiro and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39(7) (2011), 2338-2348.
• C. Wickham, Classi cation of rings with genus one zero-divisor graphs, Comm. Algebra, 36(2) (2008), 325-345.
• M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl., 11(6) (2012), 1250114 (14 pp).
• M. Ye, T. Wu, Q. Liu and J. Guo, Graph properties of co-maximal ideal graphs of commutative rings, J. Algebra Appl., 14(3) (2015), 1550027 (13 pp).
Konular Makaleler Yazar: Alpesh M. Dhorajia
 Bibtex @araştırma makalesi { ieja373659, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2018}, volume = {23}, pages = {157 - 166}, doi = {10.24330/ieja.373659}, title = {Some graph on prime ideals of a commutative ring}, key = {cite}, author = {Dhorajia, Alpesh M.} } APA Dhorajia, A . (2018). Some graph on prime ideals of a commutative ring. International Electronic Journal of Algebra, 23 (23), 157-166. DOI: 10.24330/ieja.373659 MLA Dhorajia, A . "Some graph on prime ideals of a commutative ring". International Electronic Journal of Algebra 23 (2018): 157-166 Chicago Dhorajia, A . "Some graph on prime ideals of a commutative ring". International Electronic Journal of Algebra 23 (2018): 157-166 RIS TY - JOUR T1 - Some graph on prime ideals of a commutative ring AU - Alpesh M. Dhorajia Y1 - 2018 PY - 2018 N1 - doi: 10.24330/ieja.373659 DO - 10.24330/ieja.373659 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 157 EP - 166 VL - 23 IS - 23 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.373659 UR - http://dx.doi.org/10.24330/ieja.373659 Y2 - 2019 ER - EndNote %0 International Electronic Journal of Algebra Some graph on prime ideals of a commutative ring %A Alpesh M. Dhorajia %T Some graph on prime ideals of a commutative ring %D 2018 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 23 %N 23 %R doi: 10.24330/ieja.373659 %U 10.24330/ieja.373659 ISNAD Dhorajia, Alpesh M. . "Some graph on prime ideals of a commutative ring". International Electronic Journal of Algebra 23 / 23 (Ocak 2018): 157-166. http://dx.doi.org/10.24330/ieja.373659