The super socle of the ring of continuous functions
-
Sahar Ghasemzadeh
,
Omid A. S. Karamzadeh
Abstract
We introduce and study the concept of the super socle of C(X), denoted by SCF(X) (i.e., the set of elements of C(X), which are zero everywhere except on a countable number of points of X). Using this concept we extend some of the basic results concerning CF(X), the socle of C(X), to SCF(X). In particular, we determine spaces X such that CF(X) and SCF(X) coincide. Spaces X such that Ann(SCF(X)) is generated by an idempotent are fully characterized. It is shown that SCF(X) is an essential ideal in C(X) if and only if the set of countably isolated points (i.e., points with countable neighborhoods) of X is dense in X. The one-point Lindelöffication of uncountable discrete spaces is algebraically characterized via the concept of the super socle. Consequently, it is observed that whenever Ox ⊆ SCF(X) and SCF(X) is a regular ideal (von Neumann), then X is either a countable discrete space or the one-point Lindelöffication of an uncountable discrete space. Consequently, in this case SCF(X) is a prime ideal in C(X) (note, CF(X) is never prime C(X))
Acknowledgement
We would like to thank the well-informed and a meticulous referee whose detailed report containing several constructive suggestions has greatly improved the presentation of this article.
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© 2017 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Outer measure on effect algebras
- Hamiltonian ordered algebras and congruence extension
- Automorphism groups with some finiteness conditions
- Only finitely many Tribonacci Diophantine triples exist
- Abundant semigroups with a *-normal idempotent
- Stability results for fractional differential equations with state-dependent delay and not instantaneous impulses
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- M-Cantorvals of Ferens type
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- Positive solutions of perturbed nonlinear hammerstein integral equation
- Ricci solitons on 3-dimensional cosymplectic manifolds
- Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups
- The super socle of the ring of continuous functions
- On the internal approach to differential equations 2. The controllability structure
- Finiteness of the discrete spectrum in a three-body system with point interaction
- On a subclass of Bazilevic functions
