Maximal ideals in rings of real measurable functions

Let $M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of $M(X)$, mostly in terms of certain lattice-theoretic properties of $\mathscr{A}$. The notion of $T$-measurable space is introduced. Next, we show that for every measurable space $(X,\mathscr{A})$ there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M(X)\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prove that if $(X, \mathscr{A})$ and $(Y, \mathfrak{M^{\prime}})$ are two compact $T$-measurable spaces, then $X\cong Y$ as measurable spaces if and only if $M(X)\cong M (Y)$ as rings.