#### Glasnik Matematicki, Vol. 44, No.1 (2009), 141-153.

### AN IDEAL-BASED ZERO-DIVISOR GRAPH OF A
COMMUTATIVE SEMIRING

### Shahabaddin Ebrahimi Atani

Department of Mathematics, University of Guilan, P.O. Box 1914,
Rasht, Iran

**Abstract.** There is a natural graph associated to the zero-divisors
of a commutative semiring with non-zero identity. In this article we
essentially study zero-divisor graphs with respect to primal and non-primal
ideals of a commutative semiring *R* and investigate the interplay between
the semiring-theoretic properties of *R* and the graph-theoretic properties
of Γ_{I}(*R*) for some ideal *I* of *R*. We also show that the
zero-divisor graph with respect to primal ideals commutes by the semiring
of fractions of *R*.

**2000 Mathematics Subject Classification.**
16Y60, 05C75.

**Key words and phrases.** Semiring, zero-divisor, graph, primal, ideal-based.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.44.1.07

**References:**

- D. F. Anderson and P. S. Livingston,
*The zero-divisor graph
of a commutative rings*, J. Algebra **217** (1999), 434-447.

MathSciNet
CrossRef

- P. Allen, Ideal theory in semirings, Dissertation, Texas
Christian University, 1967.

- L. Dancheng and W. Tongsuo,
*On ideal-based zero-divisor graphs*,
preprint, 2006.

- S. Ebrahimi Atani,
*On primal and weakly primal ideals over
commutative semirings*, Glas. Mat. Ser. III **43(63)** (2008), 13-23.

MathSciNet
CrossRef

- S. Ebrahimi Atani,
*The ideal theory in quotients of commutative
semirings*, Glas. Mat. Ser. III **42(62)** (2007), 301-308.

MathSciNet
CrossRef

- S. Ebrahimi Atani,
*The zero-divisor graph with respect to ideals
of a commutative semirings*, Glas. Mat. Ser. III **43(63)** (2008), 309-320.

MathSciNet
CrossRef

- S. Ebrahimi Atani and A. Yousefian Darani,
*Zero-divisor graphs
with respect to primal and weakly primal ideals*, J. Korean Math. Soc.,
**46** (2009), 313-325.

- L. Fuchs,
*On primal ideals*, Proc. Amer. Math. Soc. **1** (1950),
1-6.

MathSciNet
CrossRef

- V. Gupta and J. N. Chaudhari,
*Right π-regular semirings*, Sarajevo J. Math. **2** (2006), 3-9.

MathSciNet

- J. A. Huckaba,
Commutative rings with zero divisors, Marcel Dekker, Inc., New York, 1988.

MathSciNet

- T. G. Lucas,
*The diameter of a zero-divisor graph*, J. Algebra
**301** (2006), 174-193.

MathSciNet
CrossRef

- J. R. Mosher,
*Generalized quotients of hemirings*, Compositio
Math. **22** (1970), 275-281.

MathSciNet

- S. P. Redmond,
*An ideal-based zero-divisor graph of a commutative
ring*, Comm. Algebra **31** (2003), 4425-4443.

MathSciNet
CrossRef

*Glasnik Matematicki* Home Page