Unions of minimal prime ideals in rings of continuous functions on compact spaces

2009 ◽  
Vol 62 (2-3) ◽  
pp. 239-246 ◽  
Author(s):  
Bikram Banerjee ◽  
Swapan Kumar Ghosh ◽  
Melvin Henriksen
Keyword(s):  
Continuous Functions ◽  
Prime Ideals ◽  
Compact Spaces ◽  
Rings Of Continuous Functions ◽  
Minimal Prime

scholarly journals THE KERNELS AND CONTINUITY IDEALS OF HOMOMORPHISMS FROM 𝒞0(Ω)

2010 ◽  
Vol 88 (1) ◽  
pp. 103-130 ◽  
Author(s):  
HUNG LE PHAM
Keyword(s):  
Banach Algebras ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces

AbstractWe give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions.

Compact spaces of minimal prime ideals

1969 ◽  
Vol 111 (2) ◽  
pp. 151-158 ◽  
Author(s):  
Joseph Kist
Keyword(s):  
Prime Ideals ◽  
Compact Spaces ◽  
Minimal Prime

scholarly journals Intersections of Primary Ideals in Rings of Continuous Functions

1972 ◽  
Vol 24 (3) ◽  
pp. 502-519 ◽  
Author(s):  
R. Douglas Williams
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Necessary And Sufficient ◽  
Primary Ideals

Let C be the ring of all real valued continuous functions on a completely regular topological space. This paper is an investigation of the ideals of C that are intersections of prime or of primary ideals.C. W. Kohls has analyzed the prime ideals of C in [3 ; 4] and the primary ideals of C in [5]. He showed that these ideals are absolutely convex. (An ideal I of C is called absolutely convex if |f| ≦ |g| and g ∈ I imply that f ∈ I.) It follows that any intersection of prime or of primary ideals is absolutely convex. We consider here the problem of finding a necessary and sufficient condition for an absolutely convex ideal I of C to be an intersection of prime ideals and the problem of finding a necessary and sufficient condition for I to be an intersection of primary ideals.

scholarly journals A class of ideals in intermediate rings of continuous functions

2019 ◽  
Vol 20 (1) ◽  
pp. 109 ◽  
Author(s):  
Sagarmoy Bag ◽  
Sudip Kumar Acharyya ◽  
Dhananjoy Mandal
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Alternative Proof ◽  
Prime Ideals ◽  
Borel Sets ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Intermediate Ring ◽  
The Family ◽  
Established Fact

<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>

Transferring Results From Rings of Continuous Functions to Rings of Analytic Functions

1975 ◽  
Vol 27 (1) ◽  
pp. 75-87 ◽  
Author(s):  
Andrew Adler ◽  
R. Douglas Williams
Keyword(s):  
Entire Functions ◽  
Continuous Functions ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Completely Regular ◽  
Rings Of Continuous Functions ◽  
Maximal Ideals ◽  
Primary Ideals ◽  
Special Case ◽  
The Ideal

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.

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