اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIdeals in Rings and Intermediate Rings of Measurable Functions
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The set of all maximal ideals of the ring $mathcal{M}(X,mathcal{A})$ of real valued measurable functions on a measurable space $(X,mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $mathcal{M}(X,mathcal{A})$ in terms of the points of $hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $mathcal{M}(X,mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = mathcal{M}(X,mathcal{A})$ where $mathcal{A}$ is the $sigma$-algebra consisting of the zero-sets of $X$.
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A proper ideal $I$ in a commutative ring with unity is called a $z^circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a completely $F$-r
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