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Algebraic Properties of Certain Rings of Continuous Functions

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Algebraic Properties of Certain Rings of Continuous Functions Pacific Journal of Mathematics ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS LI PI SU Vol. 27, No. 1 January 1968 PACIFIC JOURNAL OF MATHEMATICS Vol. 27, No. 1, 1968 ALGEBRAIC PROPERTIES OF CERTAIN RINGS OF CONTINUOUS FUNCTIONS Li PI SU n f f Let X and Fbe any subsets of E , and (X , dj and (Y , d2) be any metric spaces. Let O(X), 0 ^ m ^ oo, denote the ring r of m-differentiable functions on X, and Lc(X ) be the ring of the functions which are Lipschitzian on each compact subset of X', and L(Xf) be the ring of the bounded Lipschitzian functions onl'. The relations between algebraic properties m of C (X), (resp. Le(X') or L(X) and the topological properties of X (resp. X') are studied. It is proved that if X and Y, (resp. f 1 (X , dί) and (Y , d2)) are m-realcompact, (resp. Lc-real-compact m or compact) then O(X) = C {Y) (resp. Le(X') = LC(Y>) or L(Xf) = L(Y') if and only if X and Y are O-diffeomorphic 1 (resp. (X ', di) and (Y', d2) are Lc or L-homeomorphic). During the last twenty years, the relations between the algebraic properties of Cm(X) and Cm(Y) and the topological properties of X and Y have been investigated by Hewitt [4], Myers [9], Pursell [11], Nakai [10], and Gillman and Jerison [3], where m is a positive integer, zero or infinite. In 1963, Sherbert [12] studied the ring L(X). Recently, Magill, [6] has obtained the algebraic condition relating C(X) and C(Y) (i. e., m = 0) which are both necessary and sufficient for embedding Y in X, where X and Y are two realcompact spaces. This work is to utilize the method of Gillman and Jerison [3] for m studying the algebraic properties of C (X) and Le(Xj) (§§ 2-5), and how they are related with topological properties of X and X1 respec- tively. In view of [8, Cor. 1.32], we will restrict X in Cm(X) to a m subset of E*. The results of Magill are also true in C (X) and LC(X) with some modification. In the last section, § 6, we observe some other cases. 2* Rings and ideals* Let X be an arbitrary subset of En, an ^-dimensional euclidean space, and Cm(X) be the set of all real-valued functions of class Cm in the sense of Whitney [14, § 3], where m will always refer to an arbitrary integer such that 0 ^ m <^ oo. By [15, Th. 4], we know that Cm(X) forms a ring with the identity u, the constant function of value 1, and zero element θ, the constant function of value 0. Let Cm*(X) = {feCm(X) :f is bounded}. It is clear that Cm*(X) is a subring of Cm(X) with u and θ. Let X be a metric space, and LC(X) be the set of all real-valued functions satisfy- ing Lipschitz condition on each compact subset of X [2, p. 354], We can easily show that LC(X) is a ring with u and θ. Let L(X) = 175 176 LI PI SU {feLc(X):f is bounded and Lipschitzian on entire X}, L*(X) = {feLe(X):f is bounded}. Then, both L(X) and L*(X) are the sub- rings of LC(X) with u and θ. m W Since the properties of C (X) (resp. C *(X)) and those of LC(X) (resp. Lf(X) and L(X)) are almost all the same, we will use 11 and W m 21' to denote C (X) (resp. LC(X)) and C (Γ) (resp. Lβ(Γ)), and S3 m W and 33' to denote C *(X) (resp. LC*(X), and L(X)) and C *(Γ) (resp. L?(Y) and L(Γ)) respectively, where X, and F are appropriately the subsets of En or metric space. Also " α-" and " 6-" will mean m- m m (or C -) (resp. Lc) and C *- (resp. L* and L) respectively according m W as 2ί is C (X) (resp. LC(X)) and 93 is C *(X) (resp. LC*(X), and The unit element of an / e Si or 23 is defined as usual. For /e2I, Z(f) = (XGI:/(X) = 0} is said to be the zero-set of /. Z(Ά) = {Z(f):fe$ί}. It is then clear that fell is a unit if and only if Z(f) = 0. (For Cm(X) see [15, Th. 4].) Likewise, if /GS is a unit, then Z(f) — 0. But the converse need not hold, for the multi- plicative inverse 1// of / in SI may not be a bounded function. For example: let X = E\ and f(x) = e~*eCm*(Eι) and Z{f) = 0. But 1//- e'WC^iE1). A ^-filter of Z(SI) is the same as in [3,2.2]. It is obvious that Z[I] = {Z(/) :/e J} is a 2-filter on J if 7 is a proper ideal in 21, and Z-%^~\ = {fe 3ί : Z(f) e ^} is a proper ideal if ^"is a ^-filter on X Note that it may be false that a proper ideal / c 93 implies that Z[I] is a ^-filter. For example: let us consider 33 = Cm*{Eι) and let f(x) = 1/(1 + or2), and / = (/) be the ideal generated by / in 33. Then it is clear that 0 e Z[/]. Hereafter, we will always use " ideal " to mean the proper ideal, unless the contrary is mentioned. Accordingly, every ^-filter is of the form Z[I], for some ideal I in 2ί. That Z~ι[Z[I]\ z) I is also clear. The inclusion may be proper. For instance, consider 21 = Cm(Eι). (a) For any positive integer m, let i(x) = x for all xe E\ and / = (i). Then m Z-\Z{1\\ = M0 = {fe C (E): /(0) = 0} . (Sw+1)/3 1/χ2 However, i e Mo - /. (b) In case m = oo, let /x(^) - e~ for 1 ι xeE and Λ = (/x). Then MQ = Z~[Z[I]\ contains an element ig/lβ Note that Mo is a maximal fixed ideal. Now, as for LC(X), we may consider (X, d) to be a bounded metric space, and fo(x) = (fv{x)f = (d(p,x))\ Then/OGLC(X). Let IO = (/O). Then ι Z-[Z[Io]] = {/ G LC(X): f(V) = 0} = Mv is clear. However, /p(a?) = d(p, x)e Mp — Io. ALGEBRAIC PROPERTIES OF CERTAIN RINGS 177 A 2-ultrafilter on X is a maximal z-filter [3,2.5]. We know that every subfamily of Z(St) with the finite intersection property, by Zorn's Lemma, is contained in some z-ultrafilter on X. The proofs of following propositions are obvious. PROPOSITION 2.1. If M is a maximal ideal in Sί, then Z[M] is a z-ultrafilter on X. PROPOSITION 2.2. If jy is a z-ultrafilter on X, then Z~\s^f\ is a maximal ideal in St. It follows from Propositions (2.1) and (2.2) that the mapping Z is one-one from the set of all maximal ideals in 31 onto the set of all 2-ultrafilters on X. PROPOSITION 2.3. Let M be a maximal ideal in 2ί. If Z(f) meets every member of Z[M], then feM. PROPOSITION 2.4. Let s/ be a z-ultrafilter on X. If a zero-set Z meets every member of s^f, then Z e An ideal / in SI is z-ideal if Z(f)eZ[I] implies fel. That is, / = Z~-ι[Z[I]\, [3,2.7]. It is obvious that every maximal ideal is a z- ideal. A prime ideal is defined in the usual sense. The following theorem is only true for LC(X), L?(X) or L(X). For we can show that these are lattice-ordered rings; while Cm(X) and Cm*(X) are not. THEOREM 2.5. For any z-ideal I in LC(X) (Lf(X) or L(X)) the following are equivalent: (1) I is prime. ( 2 ) / contains a prime ideal. ( 3 ) For all g,he LC(X) {Lf{X) or L(X)), g h = θ, then gel or he I. (4) For every feLc(X) (L*(X) or L(X)), there is a zero-set in Z[I] on which f does not change sign. Proof is similar to [3,2.9], 3* Zero-set, α-completely regular and α-normal spaces* We know from the proof of Lemma 25 [16, p. 669] that each closed sub- set F of En, there is an feCm(X) such that Z(f) = F. PROPOSITION 3.1. For each closed subset A of (X, d), there is feLc(X) (in fact feL(X)) such that Z(f) = A. 178 LI PI SU Proof. Let g(x) = d(A,x) and f=gΛ u\ Then /eL(X) and Z(f) = A. DEFINITION 3.2. Let X be a topological space. X is said to be α-completely regular if and only if for each closed subset F of X and x$F, there is an fe 21 such that f(x) = 1, and /[.F] = {0}. THEOREM 3.3. A topological space is a-completely regular if and only if the family Z(Ά) = {Z(f) :/e2ϊ} is a base for the closed subsets of X. Proof is similar to [3, 3.2]. DEFINITION 3.4. A topological space is said to be α-normal if for any disjoint closed subsets Fλ and F2, there is an fe% such that /[FJ = {0} and f[F2] = {1}.
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