The infinite intersection of essential ideals in any ring may not
be an essential ideal, this intersection may even be zero. By the
topological characterization of the socle by Karamzadeh and
Rostami (Proc. Amer. Math. Soc. 93 (1985), 179184), and the
topological characterization of essential ideals in Proposition
2.1, it is easy to see that every intersection of essential ideals
of C(X) is an essential ideal if and only if the set of isolated
points of X is dense in X. Motivated by this result in C(X),
we study the essentiality of the intersection of essential ideals
for topological spaces which may have no isolated points. In
particular, some important ideals C_{K}(X) and C_{∞}(X),
which are the intersection of essential ideals, are studied
further and their essentiality is characterized. Finally a
question raised by Karamzadeh and Rostami, namely when the socle
of C(X) and the ideal of C_{K}(X) coincide, is answered.
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