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The super socle of the ring of continuous functions

In: Mathematica Slovaca, vol. 67, no. 4
Sahar Ghasemzadeh - Omid A. S. Karamzadeh - Mehrdad Namdari
Detaily:
Rok, strany: 2017, 1001 - 1010
Kľúčové slová:
super socle of $C(X)$, countably isolated point, countably discrete space, c-disjoint spaces, pseudo minimal ideals, one-point Lindelöffication
O článku:
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 $O_x\subseteq SC_F(X)$ and $SC_F(X)$ 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 $SC_{F}(X)$ is a prime ideal in $C(X)$ (note, $C_{F}(X)$ is never prime $C(X)$)
Ako citovať:
ISO 690:
Ghasemzadeh, S., Karamzadeh, O., Namdari, M. 2017. The super socle of the ring of continuous functions. In Mathematica Slovaca, vol. 67, no.4, pp. 1001-1010. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0028

APA:
Ghasemzadeh, S., Karamzadeh, O., Namdari, M. (2017). The super socle of the ring of continuous functions. Mathematica Slovaca, 67(4), 1001-1010. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0028
O vydaní:
Publikované: 28. 8. 2017