پیوندهای مفید
اخبار و اعلانات
آمار
| تعداد نشریات | 43 |
| تعداد شماره‌ها | 1,744 |
| تعداد مقالات | 14,236 |
| تعداد مشاهده مقاله | 35,191,518 |
| تعداد دریافت فایل اصل مقاله | 13,977,563 |
Annihilating submodule graph for modules | ||
| Transactions on Combinatorics | ||
| مقاله 1، دوره 7، شماره 1، خرداد 2018، صفحه 1-12 اصل مقاله (269.83 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.21462 | ||
| نویسنده | ||
| Saeed Safaeeyan* | ||
| Department of mathematical Sciences, Yasouj university,Yasouj, 75918-74831, IRAN. | ||
| چکیده | ||
| Let $R$ be a commutative ring and $M$ an $R$-module. In this article, we introduce a new generalization of the annihilating-ideal graph of commutative rings to modules. The annihilating submodule graph of $M$, denoted by $\Bbb G(M)$, is an undirected graph with vertex set $\Bbb A^*(M)$ and two distinct elements $N$ and $K$ of $\Bbb A^*(M)$ are adjacent if $N*K=0$. In this paper we show that $\Bbb G(M)$ is a connected graph, ${\rm diam}(\Bbb G(M))\leq 3$, and ${\rm gr}(\Bbb G(M))\leq 4$ if $\Bbb G(M)$ contains a cycle. Moreover, $\Bbb G(M)$ is an empty graph if and only if ${\rm ann}(M)$ is a prime ideal of $R$ and $\Bbb A^*(M)\neq \Bbb S(M)\setminus \{0\}$ if and only if $M$ is a uniform $R$-module, ${\rm ann}(M)$ is a semi-prime ideal of $R$ and $\Bbb A^*(M)\neq \Bbb S(M)\setminus \{0\}$. Furthermore, $R$ is a field if and only if $\Bbb G(M)$ is a complete graph, for every $M\in R-{\rm Mod}$. If $R$ is a domain, for every divisible module $M\in R-{\rm Mod}$, $\Bbb G(M)$ is a complete graph with $\Bbb A^*(M)=\Bbb S(M)\setminus \{0\}$. Among other things, the properties of a reduced $R$-module $M$ are investigated when $\Bbb G(M)$ is a bipartite graph. | ||
| کلیدواژه‌ها | ||
| Module؛ Annihilating submodule graph؛ Complete graph | ||
| مراجع | ||
|
[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, University of Oxford, Addison-Wesley publishing company, 1969.
[2] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloq., 21 (2014) 249–256. [3] S. Akbari and A. Mohammadian, On zero-divisor graphs of finite rings, J. Algebra, 314 (2007) 168–184.
[4] D. F. Anderson, M. C. Axtell and J. A. Stickles, Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives, (M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson, Eds.), Springer-Verlag, New York, 2011.
[5] F. W. Anderson and K. R. Fuller, Ring and Category of Modules, New York, Springer-Verlag, 1992.
[6] D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring II, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001 61–72.
[7] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434–447.
[8] D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007) 543–550.
[9] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloq., 21 (2014) 249–256. [10] D. Lu and T. Wu, On bipartite zero-divisor graphs, Discrete Math., 309 no. 4 (2009) 755–762.
[11] B. Allen, E. Martin, E. New and D. Skabelund, Diameter, girth and cut vertices of the graph of equivalence classes of zero-divisors, 5 (2012) 51–60.
[12] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208–226.
[13] M. Baziar, E. Momtahan and S. Safaeeyan, A Zero-divisor Graph for Module with Respect to Their (First) Dual, J. Algebra Appl., 12 (2013) pp. 11.
[14] M. Baziar, E. Momtahan, S. Safaeeyan and N. Ranjbar, Zero-divisor graph of abelian groups, J. Algebra Appl., 13 (2014) pp. 13.
[15] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl., 10 (2011) 727–739.
[16] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl., 10 (2011) 741–753.
[17] S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (2002) 3533–3558.
[18] S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc., 51 (2014) 87–98.
[19] S. Spiroff and C. Wickham, A Zero Divisor Graph Determined by Equivalenc Classes of Zero Divisors, comm. Algebra, 39 (2011) 2338–2348.
[20] T. Y. Lam, A first Course in Noncommutative Rins, Graduate Texts in Mathematics, 131, New York, Berlin, Springer-Verlag, (1991). [21] T. Y. Lam , Lectures on Modules and Rings, Graduate Texts in Mathematics, 139, New York, Berlin, Springer-Verlag, (1998).
[22] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River, 2001. | ||
|
آمار تعداد مشاهده مقاله: 2,421 تعداد دریافت فایل اصل مقاله: 824 |
||