**April 20, 2023**. Read More

10.5120/15286-3986 |

Eswara D Rao and D Bharathi. Article: Total Graphs of Idealization. *International Journal of Computer Applications* 87(15):31-34, February 2014. Full text available. BibTeX

@article{key:article, author = {D. Eswara Rao and D. Bharathi}, title = {Article: Total Graphs of Idealization}, journal = {International Journal of Computer Applications}, year = {2014}, volume = {87}, number = {15}, pages = {31-34}, month = {February}, note = {Full text available} }

### Abstract

Let R be a commutative ring with Z(R), its set of zero divisors. The total zero divisor graph of R, denoted Z(?(R)) is the undirected (simple) graph with vertices Z(R)*=Z(R)-{0}, the set of nonzero zero divisors of R. and for distinct x, y ? z(R)*, the vertices x and y are adjacent if and only if x + y ? z(R). In this paper prove that let R is commutative ring such that Z(R) is not ideal of R then Z(?(R(+)M)) is connected with diam(Z(?(R(+)M))) = 2 and the sub graphs Z(?(R(+)M)) and Reg(?(R(+)M)) of T(?(R(+)M)) are not disjoint. And also prove that let R be a commutative ring such that Z(R) is not an ideal of R with Z(R(+)M) = Z(R)(+)M and Reg(R(+)M) = Reg(R)(+)M then Z(?(R(+)M)) is connected if and only if Z(?(R) is connected and Reg(?(R(+)M)) is connected if and only if Reg(?(R).

### References

- I. Beck, Coloring of commutative rings, J. Algebra 116(1988),208-226.
- D. D. Anderson,M. Naseer, Becks coloring of a commutative ring, J. Algebra,159(1993), 500-514.
- D. F. Anderson,P. S. Livingston, The zero divisor graph of a commutative ring J. Algebra, 217(1999),434-447.
- D. F. Anderson, A. Frazier,A. Lauve, P. Livingston ,The zero divisor graph of a commutative ring II , Lecture Notes in Pure and Appl. Math. ,vol. 220,Dekker,New York,2001,61-72.
- D. F. Anderson,A. Badawi, The total graph of commutative ring, J. Algebra,. 320, 2706-2719,(2008)
- F. R. DEMEYER,T. McKenzie and K. Schneider,the zero divisor graph of a commutative semigroup. Semigroup Forum,vol. 65(2002),206-214.
- D. F. Anderson, A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36 (2008) 3073–3092.
- D. F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003) 221–241.
- D. F. Anderson, S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra 210 (2007) 543–550.
- M. Axtel, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005) 2043–2050.
- M. Axtel, J. Stickles, Zero-divisor graphs of idealizations, J. Pure Appl. Algebra 204 (2006) 235–243.
- I. Kaplansky, Commutative Rings, rev. ed. , University of Chicago Press, Chicago, 1974.
- J. D. LaGrange, Complemented zero-divisor graphs and Boolean rings, J. Algebra 315 (2007) 600–611.
- T. G. Lucas, The diameter of a zero-divisor graph, J. Algebra 301 (2006) 174–193.
- S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring,J. Algebra 274 (2004) 847-855.