实可测函数环中的极大理想
Maximal ideals in rings of real measurable functions
摘要：
设$ M (X)$是可测空间$(X, \mathscr{A})$上所有实可测函数的环。本文证明了$M(X)$的每个理想都是$Z^{\circ}$ -理想.此外，我们给出了$M(X)$的极大理想的几个特征，主要是根据$\mathscr{A}$的某些格论性质。引入了$T$ -可测空间的概念。其次，我们证明了对于每个可测空间$(X,\mathscr{A})$，存在一个$T$ -可测空间$(Y,\mathscr{A}^{\prime})$，使得$M(X)\cong M(Y)$为环.引入了紧可测空间的概念。其次，证明了如果$(X, \mathscr{A})$和$(Y, \mathfrak{M^{\prime}})$是两个紧的$T$ -可测空间，则$X\cong Y$是可测空间当且仅当$M(X)\cong M (Y)$是环.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
