C(X) vs. C(X) modulo its socle

Abstract

Let C F (X) be the socle of C(X). It is shown that each prime ideal in C(X)/C F (X) is essential. For each h ∈ C(X), we prove that every prime ideal (resp. zideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that dim(C(X)/C F (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/C F (X) is essential in C(X)/C F (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)/C F (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. Introduction. Let C(X) be the ring of real-valued continuous functions on an infinite completely regular Hausdorff space X, and C ⋆ (X) be its subring of bounded functions. The socle of C(X), denoted by C F (X), is the sum of all minimal ideals of C(X), which is the intersection of all essential ideals in C(X) (recall that, an ideal is essential if it intersects every nonzero ideal nontrivially); see [15]. It can be easily seen that C F (X) = 0 if and only if X has isolated points, and when the set of isolated points in X is finite, then C(X)/C F (X) ∼ = C(Y), where Y is the set of nonisolated points of X. This implies that in this case C(X)/C F (X) ∼ = eC(X), where e 2 = e ∈ C(X), as a ring will enjoy all the general algebraic properties of C(X). Although in general C(X)/C F (X) may not be isomorphic to C(Y) for any topological space Y , in any case, one encounters a curious similarity between the two rings C(X) and C(X)/C F (X), and their common properties usually give rise to useful information about X. For example, X is a P-space (resp. an extremally disconnected P-space with only a finite number of isolated points) if and only if C(X) or equivalently C(X)/C F (X) is an ℵ 0-self-injective (resp. self-injective) ring (see [11, Theorems 1, 2 and Lemma 3.1]). Neither of the two partially ordered rings C(X) and C(X)/C F (X) (note that C F (X) is

On the essentiality and primeness of λ-super socle of C(X)

“…So each maximal ideal M containing S λ (X) is λessential, since C F (X) ⊆ S λ (X). We also recall that every prime ideal in C(X) is either essential or it is a maximal ideal which is generated by idempotent and it is a minimal prime too, see [4]. In view of these facts and using the above proposition and the fact that S λ (X) is a z-ideal (hence it is an intersection of prime ideals), we immediately have the following proposition.…”

On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

“…It is well known and easy to show that a nonzero ideal I in a reduced ring R (i.e., no nonzero element in R is nilpotent) is essential if and only if Ann(I) = 0, see [3,Background and preliminary results]. The proof of the following corollary is similar to [11,Corollary 5.4], but we include the proof for the sake of the reader.…”

“…We recall that C F (X) is never a prime ideal of C(X), see [8, Proposition 1.2], or [3,Remark 2.4]. The following result characterizes spaces X such that C F (X) = 0 is a prime ideal in L c (X) (note, C F (X) = 0 if and only if X has isolated points).…”

“…Finally, in this section similar to C(X) and C c (X), we prove that L c (X) is a regular ring if and only if it is ℵ 0 -selfinjective. The socle of C(X) (i.e., C F (X)) which is in fact a direct sum of minimal ideals of C(X) is characterized topologically in [20,Proposition 3.3], and it turns out that C F (X) is a useful object in the context of C(X), see [20], [1], [2], [8], [3], and [6]. The socle of C c (X), denoted by Soc C c (X) , is studied in [11,Proposition 5.3], and spaces X for which Soc C c (X) = C F (X) are determined in [11,Theorem 5.6].…”

“…Spaces X for which Soc L c (X) = Soc C c (X) and Soc L c (X) = C F (X) are also characterized. In [8,Proposition 1.2], [3,Remark 2.4], it is shown that C F (X) can not be a prime ideal in C(X), where X is any space. But, in [11,Proposition 6.2], spaces c AGT, UPV, 2015 Appl.…”