# On the Zero-Divisor Graph of a Ring

@article{Anderson2008OnTZ, title={On the Zero-Divisor Graph of a Ring}, author={David F. Anderson and Ayman Badawi}, journal={Communications in Algebra}, year={2008}, volume={36}, pages={3073 - 3092} }

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals…

## 126 Citations

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## References

SHOWING 1-10 OF 41 REFERENCES

### Factoring Nonnil Ideals into Prime and Invertible Ideals

- Mathematics
- 2005

For a commutative ring R, let (R) be the set of all nilpotent elements of R, Z(R) the set of all zero divisors of R, and T(R) the total quotient ring of R. Set H = {R | R is a commutative ring and…

### Beck′s Coloring of a Commutative Ring

- Mathematics
- 1993

A commutative ring R can be considered as a simple graph whose vertices are the elements of R and two different elements x and y of R are adjacent if and only if xy = 0. Beck conjectured that χ(R) =…

### On f-Prüfer Rings and f-Bezout Rings

- Mathematics
- 2004

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prufer domains and Bezout domains. Let R be a commutative ring with 1 such that Nil(R)…

### ON φ-PRÜFER RINGS AND φ-BEZOUT RINGS

- Mathematics
- 2004

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let H = {R | R is a commutative ring with 1 6= 0 and…

### On divided commutative rings

- Mathematics
- 1999

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided,…