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Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|<\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.
We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$.
Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective.
We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$
are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$.
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Abstract
Let Cc(X) (resp. CF(X)) denote the subring of C(X) consisting of functions with countable (resp. finite) image and CF(X) be the socle of C(X). We characterize spaces X with C∗(X)=Cc(X), which generalizes a celebrated result due to Rudin, Pelczynnski and Semadeni. Two zero-dimensional compact spaces X, Y are homeomorphic if and only if Cc(X)≅Cc(Y) (resp. CF(X)≅ CF(Y)). The spaces XX for which Cc(X)=CF(X) are characterized. The socles of Cc(X), CF(X), which are observed to be the same, are topologically characterized and spaces XX for which this socle coincides with CF(X) are determined, too. A certain well-known algebraic property of C(X), where X is realcompact, is extended to Cc(X). In contrast to the fact that CF(X) is never prime in C(X), we characterize spaces X for which CF(X) is a prime ideal in Cc(X). It is observed for these spaces, Cc(X) coincides with its own socle (a fact, which is never true for C(X), unless X is finite). Finally, we show that a space X is the one-point compactification of a discrete space if and only if CF(X) is a unique proper essential ideal in CF(X).

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))

The concept of λ-super socle of C(X), denoted by Sλ(X) (i.e., the set of elements of C(X) such that the cardinality of their cozerosets are less than λ, where λ is a regular cardinal number with λ≤|X|) is introduced and studied. Using this concept we extend some of the basic results concerning SCF(X), the super socle of C(X) to Sλ(X), where λ≥ℵ0. In particular, we determine spaces X for which SCF(X) and Sλ(X) coincide. The one-point λ-compactification of a discrete space is algebraically characterized via the concept of λ-super socle. In fact we show that X is the one-point λ-compactification of a discrete space Y if and only if Sλ(X) is a regular ideal and Sλ(X)=Ox, for some x∈X.
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Let CF(X) denote the socle of C(X). It is shown that X is a P-space if and only if C(X) is a א0-selnnjective ring or equivalently, if and only if C(X)/CF(X) is א0-selfmjective. We also prove that X is an extremally disconnected P-space with only a finite number of isolated points if and only if C(X)/CF(X) is selfinjective. Consequently, if X is a P-space, then X is either an extremally disconnected space with at most a countable number of isolated points or both C(X) and C(X)/CF(X) have uncountable Goldie-dimensions. Prime ideals of C(X)/CF(X) are also studied.

Not every ring homomorphism contracts the socle of its codomain to an ideal contained in the socle of its domain. Rarer still does a homomorphism contract the socle to the socle. We find conditions on a frame homomorphism that ensure that the induced ring homomorphism contracts the socle of the codomain to an ideal contained in the socle of the domain. The surjective among these frame homomorphisms induce ring homomorphisms that contract the socle to the socle. These homomorphisms characterize P -frames as those L for which every frame homomorphism with L as domain is of this kind.

The above topology is defined and studied on C(X), the ring of real-valued continuous functions on a completely regular Hausdorff space X. The minimal ideals and the socle of C(X) are characterized via their corresponding z-filters. We observe that these ideals are z-ideals and X is discrete if and only if the socle of C(X) is a free ideal. It is also shown that for a class of topological spaces, containing all P-spaces, the family Ck(X) of functions with compact support is identical with the intersection of the free maximal ideals of C(X).

Abstract. Let CF (X) be the socle of C(X). It is shown that each prime ideal in
C(X)/CF (X) is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated
points (resp. intZ(h) = ∅). It is proved that dim(C(X)/CF (X)) ≥ dim C(X), where
dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We
also give an algebraic characterization of compact spaces with at most a countable number
of nonisolated points. For each essential ideal E in C(X), we observe that E/CF (X) is
essential in C(X)/CF (X) if and only if the set of isolated points of X is finite. Finally,
we characterize topological spaces X for which the Jacobson radical of C(X)/CF (X)
is zero, and as a consequence, we observe that the cardinality of a discrete space X is
nonmeasurable if and only if υX, the realcompactification of X, is first countable

Let Cc(X) = {f ∈ C(X):f(X)is countable}. Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring Cc(X) is either Cc(X) or a semiprime (resp.prime) ideal in Cc(X). For an ideal I in Cc(X), it is observed that I and √I have the same largest zc-ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that Cc(X) ≅ Cc(Y). Consequently, if X has only countable number of components, then Cc(X) ≅ C(Y) for some zero-dimensional space Y. Spaces X for which Cc(X) is regular (called CP-spaces) are characterized both algebraically and topologically and it is shown that P-spaces and CP-spaces coincide when X is zero-dimensional. In contrast to C*(X), we observe that Cc(X) enjoys the algebraic properties of regularity, No-selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that Cc(X) is not isomorphic to any C(Y) is given.

In this paper, we present a new subring of C(X) that contains the subring Cc(X), the set of all continuous functions with countable image. Let Lcc(X) = {f ∈ C(X): |X \Cf ≤ ℵ0}, where Cf is the union of all open subsets U ⊆ X such that |f(U)| ≤ ℵ0. We observe that Lcc(X) enjoys most of the important properties which are shared by C(X) and Cc(X). It is shown that any hereditary lindelöf scattered space is functionally countable. Spaces X such that Lcc(X) is regular (von Neumann) are characterized and it is shown that ℵ0-selfinjectivity and regularity of Lcc(X) coincide.

- K R Goodearl

Goodearl, K. R. Von Neumann Regular Rings (Pitman, 1979).

- M Namdari
- S Soltanpour

Namdari, M. and Soltanpour, S. Locally Socle Of C(X), JAMM. 4 (2), 87-99, 2014.