Document Type : Original Manuscript


1 Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran.

2 Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, City, Country.


Let $M(X, \mathcal{A}, \mu)$ be the ring of real-valued measurable functions
on a measurable space $(X, \mathcal{A})$ with measure $\mu$.
In this paper, we study the zero-divisor graph of $M(X, \mathcal{A}, \mu)$,
denoted by $\Gamma(M(X, \mathcal{A}, \mu))$.
We give the relationships among graph properties of $\Gamma(M(X, \mathcal{A}, \mu))$, ring properties of
$M(X, \mathcal{A}, \mu)$ and measure properties of $(X, \mathcal{A}, \mu)$.
Finally, we investigate the continuity properties of $\Gamma(M(X, \mathcal{A}, \mu))$.


1. M. Abedi, Zero-divisor graph of real-valued continuous functions on a frame, Filomat, 33(1) (2019), 135–146.
2. S. Acharyya, S. K. Acharyya, S. Bag and J. Sack, Recent progress in rings and subrings of real valued measurable functions, Quaest. Math., 43(7) (2019), 959–
3. S. K. Acharyya, S. Bag and J. Sack, Ideals in rings and intermediate rings of measurable functions, J. Algebra Appl., 19(2) (2020), 205–238.
4. A. Amini, B. Amini, E. Momtahan and M. H. Shirdeh Haghighi, Generalized rings of measurable and continuous functions, Bull. Iranian Math. Soc., 39(1) (2013), 49–64.
5. D. F. Anderson, A. Frazier, A. Lauve and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.
6. D. F. Anderson and G. D. LaGrange, Some remarks on the compressed zerodivisor graph, J. Algebra, 447 (2016), 297–321.
7. H. Azadi, M. Henriksen and E. Momtahen, Some properties of algebras of realvalued mesurable functions, Acta Math. Hungar., 124(1-2) (2009), 15–23.
8. F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hungar.,
108(1-2) (2005), 25–36.
9. I. Beck, Coloring of commutative rings, J. Algebra, 11 (1988), 208–2267.
10. J. Connor and E. Avas, Lacunary statical and sliding window convergence for measurable functions, Acta Math. Hungar., 145(2) (2015), 416–4327.
11. R. Engelking, General topology, Heldermann Verlag, Berlin, 1989.
12. A. A. Estaji and T. Haghdadi, Zero- divisor graph for S-act, Lobachevskii J. Math., 36(1) (2015), 1–8.
13. K. R. Goodearl, Von neumann regular rings, Krieger Publishing Company, New York, 1991.
14. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1989.
15. A. W. Hager, Algebras of measurable functions, Duke Math. J., 38 (1971),