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Register a new userIdeals 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 $F$-regular topological space $X$, let $C(X,F)$ be the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the aggregate of all those functions which are bounded over $X$. An explicit formula for all the $z^circ$-ideals in $A(X,F)$ in terms of ideals of closed sets in $X$ is given. It turns out that an intermediate ring $A(X,F) eq C(X,F)$ is never regular in the sense of Von-Neumann. This property further characterizes $C(X,F)$ amongst the intermediate rings within the class of $P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and only if each maximal ideal in $C(X,F)$ is $z^circ$-ideal. Incidentally this property also characterizes $C(X,F)$ amongst the intermediate rings within the family of almost $P_F$-spaces.
Two separated realcompact measurable spaces $(X,mathcal{A})$ and $(Y,mathcal{B})$ are shown to be isomorphic if and only if the rings $mathcal{M}(X,mathcal{A})$ and $mathcal{M}(Y,mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $mathcal{M}(X,mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $mu$ on $(X,mathcal{A})$, the $m_mu$-topology on $mathcal{M}(X,mathcal{A})$ is 1st countable if and only if it is connected and this happens when and only when $mathcal{M}(X,mathcal{A})$ becomes identical to the subring $L^infty(mu)$ of all $mu$-essentially bounded measurable functions on $(X,mathcal{A})$. Additionally, we investigate the ideal structures in subrings of $mathcal{M}(X,mathcal{A})$ that consist of functions vanishing at all but finitely many points and functions vanishing at infinity respectively. In particular, we show that the former subring equals the intersection of all free ideals in $mathcal{M}(X,mathcal{A})$ when $(X,mathcal{A})$ is separated and $mathcal{A}$ is infinite. Assuming $(X,mathcal{A})$ is locally finite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of $mathcal{M}(X,mathcal{A})$.
A definition of quasi-flat left module is proposed and it is shown that any left module which is either quasi-projective or flat is quasi-flat. A characterization of local commutative rings for which each ideal is quasi-flat (resp. quasi-projective) is given. It is also proven that each commutative ring R whose finitely generated ideals are quasi-flat is of $lambda$-dimension $le$ 3, and this dimension $le$ 2 if R is local. This extends a former result about the class of arithmetical rings. Moreover, if R has a unique minimal prime ideal then its finitely generated ideals are quasi-projective if they are quasi-flat. In [1] Abuhlail, Jarrar and Kabbaj studied the class of commutative fqp-rings (finitely generated ideals are quasi-projective). They proved that this class of rings strictly contains the one of arithmetical rings and is strictly contained in the one of Gaussian rings. It is also shown that the property for a commutative ring to be fqp is preserved by localization. It is known that a commutative ring R is arithmetical (resp. Gaussian) if and only if R M is arithmetical (resp. Gaussian) for each maximal ideal M of R. But an example given in [6] shows that a commutative ring which is a locally fqp-ring is not necessarily a fqp-ring. So, in this cited paper the class of fqf-rings is introduced. Each local commutative fqf-ring is a fqp-ring, and a commutative ring is fqf if and only if it is locally fqf. These fqf-rings are defined in [6] without a definition of quasi-flat modules. Here we propose a definition of these modules and another definition of fqf-ring which is equivalent to the one given in [6]. We also introduce the module property of self-flatness. Each quasi-flat module is self-flat but we do not know if the converse holds. On the other hand, each flat module is quasi-flat and any finitely generated module is quasi-flat if and only if it is flat modulo its annihilator. In Section 2 we give a complete characterization of local commutative rings for which each ideal is self-flat. These rings R are fqp and their nilradical N is the subset of zerodivisors of R. In the case where R is not a chain ring for which N = N 2 and R N is not coherent every ideal is flat modulo its annihilator. Then in Section 3 we deduce that any ideal of a chain ring (valuation ring) R is quasi-projective if and only if it is almost maximal and each zerodivisor is nilpotent. This complete the results obtained by Hermann in [11] on valuation domains. In Section 4 we show that each commutative fqf-ring is of $lambda$-dimension $le$ 3. This extends the result about arithmetical rings obtained in [4]. Moreover it is shown that this $lambda$-dimension is $le$ 2 in the local case. But an example of a local Gaussian ring R of $lambda$-dimension $ge$ 3 is given.
Let $Sigma (X,mathbb{C})$ denote the collection of all the rings between $C^*(X,mathbb{C})$ and $C(X,mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^circ$-ideals in the rings $P(X,mathbb{C})$ in $Sigma(X,mathbb{C})$ and in their real-valued counterparts $P(X,mathbb{C})cap C(X)$. It is shown that the structure space of any such $P(X,mathbb{C})$ is $beta X$. We show that for any maximal ideal $M$ in $C(X,mathbb{C}), C(X,mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{mathcal{P}}(X,mathbb{C})$ of $C(X,mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
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