Maximal ideals in rings of real measurable functions

Abstract

Let M(X) be the ring of all real measurable functions on a measurable space (X,A). In this article, we show that every ideal of M(X) is a Z?-ideal. Also, we give several characterizations of maximal ideals of M(X), mostly in terms of certain lattice-theoretic properties of A. The notion of T-measurable space is introduced. Next, we show that for every measurable space (X,A) there exists a T-measurable space (Y,A') such that M(X) ? M(Y) as rings. The notion of compact measurable space is introduced. Next, we prove that if (X,A) and (Y,A') are two compact T-measurable spaces, then X ? Y as measurable spaces if and only if M(X) ? M(Y) as rings.