ZERO-DIVISOR GRAPH OF THE RINGS OF REAL MEASURABLE FUNCTIONS WITH THE MEASURES

Abstract

Let $M(X, mathcal{A}, mu)$ be the ring of real-valued measurable functionson 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 $Gamma(M(X, mathcal{A}, mu))$.