rings of continuous functions
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2021 ◽  
pp. 1-26
Author(s):  
GURAM BEZHANISHVILI ◽  
LUCA CARAI ◽  
PATRICK J. MORANDI




Author(s):  
E. Ghashghaei

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on [Formula: see text] to characterize spaces [Formula: see text] when every irreducible ideal of [Formula: see text] is pseudoprime. We also characterize the rings of continuous functions [Formula: see text] in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam.







2020 ◽  
Author(s):  
Charles E. Aull


Author(s):  
A. Khelif ◽  
D. Scarpalezos

Abstract In this paper we investigate the structure of the set of maximal ideals of $${{\mathcal {G}}}(\Omega )$$ G ( Ω ) . The method of investigation passes through the use of the $$m-$$ m - reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.



2020 ◽  
Vol 284 ◽  
pp. 107360
Author(s):  
Themba Dube ◽  
Dorca Nyamusi Stephen




2019 ◽  
Vol 20 (1) ◽  
pp. 109 ◽  
Author(s):  
Sagarmoy Bag ◽  
Sudip Kumar Acharyya ◽  
Dhananjoy Mandal

<p>For  any  completely  regular  Hausdorff  topological  space X,  an  intermediate  ring A(X) of  continuous  functions  stands  for  any  ring  lying between C<sup>∗</sup>(X) and C(X).  It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals.  It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X.  Interrelation between z-ideals, z◦-ideal and Ʒ<sub>A</sub>-ideals in A(X) are examined.  It is proved that within the family of almost P-spaces X, each Ʒ<sub>A</sub> -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).</p>



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