When is the super socle of C(X) prime?

• Autores: Sahar Ghasemzadeh, Mehrdad Namdari
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 20, Nº. 1, 2019, págs. 231-236
• Idioma: inglés
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• Resumen

  • Let $SC_F(X)$ denote the ideal of $C(X)$ consisting of functions which are zero everywhere except on a countable number of points of $X$. It is generalization of the socle of C(X) denoted by $C_F(X)$. Using this concept we extend some of the basic results concerning $C_F(X)$ to $SC_F(X)$. In particular, we characterize the spaces $X$ such that $SC_F(X)$ is a prime ideal in $C(X)$ (note, $C_F(X)$ is never prime ideal in $C(X)$). This may be considered as an advantage of $SC_F(X)$ over $C_F(X)$. We are also interested in characterizing topological spaces $X$ such that $C_c(X)=\mathbb{R}+SC_F(X)$, where $C_c(X)$ denotes the subring of $C(X)$ consisting of functions with countable image.