# Domination Number of Nagata Extension Ring

Document Type: Research Articles

Author

Department of Science , Bushehr Branch, Islamic Azad University, Bushehr, Iran

Abstract

Aََََbstract:Let R is a commutative ring whit Z(R) as the set of zero divisors. The total graph of R, denoted by T ((R)) is the (undirected) graph with all elements of R as vertices, and two distinct vertices are adjacent if their sum is a zero divisor. For a graph G = (V; E), a set S is a dominating set if every vertex in V n S is adjacent to a vertex in S. The domination number is equal |S|where |S| is minimum. For R-module M, an Nagata extension (idealization), denoted by R(+)M is a ring with identity and for two elements (r; m); (s; n) of R(+)M we have (r; m) + (s; n) = (r + s; m + n) and (r; m)(s; n) = (rs; rn + sm). In this paper, we seek to determine the bound for the domination number of total graph T ((R(+)M)).

Keywords

### References

[1]S. Akbari, D. Kiani, F. Mohammadi, S. Moradi, The total graph and
regular graph of a commutative ring, J. Pure Appl. Algebra, 2009, 213(12),
2224-2228.

[2]S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is
planar or a complete r-regular graph, J. Algebra, 2003, 270, 169-180

[3]D. F. Anderson and A. Badawi,The total graph of a commutative ring, J.
Algebra, 2008, 320, 2706-2719 Preprint.

[4]D. F. Anderson and A. Badawi, On the total graph of a commutative ring
without the zero element, J. Algebra, 2012, 11(4),

[5]D. F. Anderson and P. S. Livingston, The zero-divisor graph of a
commutative ring, J. Algebra, 1999, 217, 434-447.

[6]D. D. Anderson and Michael Winders, Idealization of a module, J.
Algebra, 2009, 1(1), 123-134.

[7]M. Axtell, J. Coykandall and J. Stickles, Zero-divisor graphs of polynomial
and power series over commutative rings, Comm. Algebra, 2005, 33, 20432050.

[8]M. Axtell and J. Stickles, Zero-divisor graphs of idealizations, J. Pure
Appl. Algebra, 2006, 204, 235-243.

[9]I. Beck, Coloring of commutative ring, J. Algebra, 1988, 116(1), 208-226.

[10]T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 2006, 301,
174-193.

[11]H. R. Maimani C. Wickham, S. Yassemi, Rings whose total graphs have
genus at most one, Rocky Mountain J. Math., 2012, 42, 1551-1560.

[12]A. Shariatinia, H. R. Maimani, S. Yassemi, Domination Number Of Total
Graphs, Mathematica Slovaca, 2016, 66(6), 34-55.