A KIND OF GRAPH STRUCTURE ASSOCIATED WITH ZERO-DIVISORS OF MONOID RINGS | ||
| Journal of Algebraic Systems | ||
| دوره 11، شماره 2، فروردین 2024، صفحه 53-63 اصل مقاله (151.48 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2022.12238.1646 | ||
| نویسندگان | ||
| Mohammad Etezadi1؛ Abdollah Alhevaz* 2 | ||
| 1Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran. | ||
| 2Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran | ||
| چکیده | ||
| Let $R$ be an associative ring and $M$ be a monoid. In this paper, we introduce new kind of graph structure asociated with zero-divisors of monoid ring $R[M]$, calling it the $M$-Armendariz graph of a ring $R$ and denoted by $A(R,M)$. It is an undirected graph whose vertices are all non-zero zero-divisors of the monoid ring $R[M]$ and two distinct vertices $\alpha=a_{1}g_{1}+\cdots+ a_{n}g_{n}$ and $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ are adjacent if and only if $a_{i}b_{j}=0$ or $b_{j}a_{i}=0$ for all $i,j$. We investigate some graph properties of $A(R,M)$ such as diameter, girth, domination number and planarity. Also, we get some relations between diameters of the $M$-Armendariz graph $A(R,M)$ and that of zero divisor graph $\Gamma(R[M])$, where $R$ is a reversible ring and $M$ is a unique product monoid. | ||
| کلیدواژهها | ||
| $M$-Armendariz graph؛ diameter؛ unique product monoid؛ monoid ring؛ domination number | ||
| مراجع | ||
|
1. C. Abdiog˘lu and E. Y. Celikel, The Armendariz graph of a ring, Discussiones Mathematicae, General Algebra and Applications, 38 (2018), 189–196.
2. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434–447.
3. I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
4. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra, 265 (2003), 457–477.
5. B. Bollobas, Modern Graph Theory, Grad. Text in Math., Vol. 184, SpringerVerlag, New York, 1998.
6. P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(6) (1999), 641–648.
7. E. Hashemi, McCoy rings relative to a monoid, Comm. Algebra, 38 (2010), ئ1075–1083.
8. E. Hashemi and A. Alhevaz, Undirected zero divisor graphs and unique product monoid rings, Algebra Colloq., 26(4) (2019), 665–676.
9. E. Hashemi and R. Amirjan, Zero-divisor graphs of Ore extensions over reversible rings. Canad. Math. Bull., 59(4) (2016), 794–805.
10. E. Hashemi, R. Amirjan and A. Alhevaz, On zero-divisor graphs of skew polynomial rings over non-commutative rings, J. Algebra Appl., 16(3) (2017) Article ID: 1750056, 14 pp.
11. C. Y. Hong, N. K. Kim, Y. Lee, and S. J. Ryu, Rings with property (A) and their extensions, J. Algebra, 315(2) (2007), 612–628.
12. Z. Liu, Armendariz ring relative to a monoid, Comm. Algebra, 33 (2005), 649– 661.
13. G. Marks, R. Mazurek and M. Ziembowski, A new class of unique product monoids with applications to ring theory, Semigroup Forum, 78 (2009), 210–225.
14. J. Okniski, Semigroup Algebra, New York: W. A. Benjamin, 1991.
15. D. S. Passman, The Algebraic structure of group rings, Wiley, New York, 1977.
16. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings, 1 (2002), 203–211 | ||
|
آمار تعداد مشاهده مقاله: 273 تعداد دریافت فایل اصل مقاله: 319 |
||