A Ring of Analytic Functions.

Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains.These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest.Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered.In §1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to ¿>X (1.2), the Boolean space of the algebra of open-and-closed sets of X.The maximal ideal space of C(X,Z)is shown to map continuously onto oX (1.3).The space, o0X, of points of ¿X that give rise to integer residue class domains, is studied in §2.The map of X into <50X strongly resembles the realcompactification injection [GJ].A representation theorem of C(X,Z) over o0X is also given (2.4).It is shown in §3 that points in SX -o0X give rise to Z, a nonstandard model of Z (3.1).Here some of the relevant background material in model theory is discussed.The algebraic theory of nonstandard arithmetic is studied in §4.In §5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q.In §6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3).The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal.The ideals of Z are analyzed in §7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4).Ideals in C(X,Z) are decomposed in §8, first into coprimary ideals (8.4), and then into primary ideals (8.9).In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X,Z) are analyzed.In §9 some model-theoretic results are obtained on the residue class fields of C(X,Z), the principal result being that any such field is elementarily equivalent Presented to the Society, January 23,1964 under the title Rings of integer-valued functions and nonstandard models; received by the editors June 12,1964.

Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains.These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest.Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered.In 1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to >X (1.2), the Boolean space of the algebra of open-and-closed sets of X.The maximal ideal space of C(X,Z)is shown to map continuously onto oX (1.3).The space, o0X, of points of X that give rise to integer residue class domains, is studied in 2.The map of X into <50X strongly resembles the realcompactification injection [GJ].A representation theorem of C(X,Z) over o0X is also given (2.4).It is shown in 3 that points in SX -o0X give rise to Z, a nonstandard model of Z (3.1).Here some of the relevant background material in model theory is discussed.The algebraic theory of nonstandard arithmetic is studied in 4.In 5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q.In 6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3).The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal.The ideals of Z are analyzed in 7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4).Ideals in C(X,Z) are decomposed in 8, first into coprimary ideals (8.4), and then into primary ideals (8.9).In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X,Z) are analyzed.In 9 some model-theoretic results are obtained on the residue class fields of C(X,Z), the principal result being that any such field is elementarily equivalent Presented to the Society, January 23,1964 under the title Rings of integer-valued functions and nonstandard models; received by the editors June 12,1964.

Invariant means and invariant ideals in L∞( G) for a locally compact group G

Left ideal structure of C∗-algebras

Let R R be a Noetherian, integrally closed local domain, and Λ \Lambda an R R -order in a generalized quaternion algebra over the quotient field of R R . In this note, it is proved that: (a) Λ \Lambda has at most two maximal ideals; and (b) in case Λ \Lambda does have exactly two maximal ideals, the corresponding residue class rings are commutative fields.

Prime and maximal ideals in subrings of C( X)

Let G denote a locally compact abelian topological group (an l.c.a.group) with dual group C\ We will denote by A7(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf.[1], [2], and [5]).In this paper we shall be concerned with closed subalgebras 501 of M(G) with the following two properties:(1) if p e 50c and v is absolutely continuous with respect to p, then vs;(2) the maximal ideal space of 501 is G~, where the Gelfand transform p~ of P e 501 is given by p~(x)=J x dp for ^Ga; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on 50.Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G).It is well known thatL(G) satisfies ( 1) and (2) (cf.[5, Chapter 1]).In Theorem 1 we show that any L-subalgebra 50c of M(G), with L(G)<=50cc(L(G))1,2) also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G).We conjecture that the converse is also true; i.e., if 501 satisfies (1) and (2) then L(G)c 50c c(L(G))1/2.In Theorem 2 we prove that this is true provided G contains no copy of 7?" for > 1.The problem arises in the following way : in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 501, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of 5D.M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup.The map p-> ps which embeds 50c in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of 50 as a space of measures.The maximal ideal space of 50c may be identified as S~, the set of all semicharacters of S, where the Gelfand transform of p e 50c is given by p~(f) = $fdp.sfor fe S~ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero).Under pointwise multiplication S~ is a semigroup provided 50c has an approximate identity.

Let G denote a locally compact abelian topological group (an l.c.a. group) with dual group G^. We will denote by M(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf. [1], [2], and [5]). In this paper we shall be concerned with closed subalgebras 91 of M(G) with the following two properties: (1) if pu E 9i and v is absolutely continuous with respect to pu, then v E W9; (2) the maximal ideal space of 91 is G', where the Gelfand transform ju^ of uE9 is given by pu^(x)= f dpu for X E G^; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on W9. Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G). It is well known that L(G) satisfies (1) and (2) (cf. [5, Chapter 1]). In Theorem 1 we show that any L-subalgebra 91 of M(G), with L(G) c 91 c (L(G))112, also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G). We conjecture that the converse is also true; i.e., if 91 satisfies (1) and (2) then L(G) c 9Nc(L(G))1 2. In Theorem 2 we prove that this is true provided G contains no copy of Rn for n > 1. The problem arises in the following way: in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 9N, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of WM. M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup. The map pu -* pu, which embeds 91 in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of St as a space of measures. The maximal ideal space of Tt may be identified as S^, the set of all semicharacters of S, where the Gelfand transform of p. E 9 is given by iu$(f)= f f dps forfe S^ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero). Under pointwise multiplication S^ is a semigroup provided 91 has an approximate identity.

Exponential algebra is a new algebraic structure consisting of a semigroup structure, a scalar multiplication, an internal multiplication and a partial order [introduced in [4]]. This structure is based on the structure `exponential vector space' which is thoroughly developed by Priti Sharma et. al. in [11] [This structure was actually proposed by S. Ganguly et. al. in [1] with the name `quasi-vector space'] Exponential algebra can be considered as an algebraic ordered extension of the concept of algebra. In the present paper we have shown that the function space $ C^+(\mathbf X) $ of all non-negative continuous functions on a topological space $\mathbf X$ is a topological exponential algebra under the compact open topology. Also we have discussed the ideals and maximal ideals of $ C^+(\mathbf X) $. We find an ideal of $ C^+(\mathbf X)$ which is not a maximal ideal in general; actually maximality of that ideal depends on the topology of $\mathbf{X}$. The concept of ideals of exponential algebra was introduced by us in [4].

Introduction.The purpose of this paper is to study the ring C(X, Z) of all integer-valued continuous functions on a topological space X.Our subject is similar in many ways to the ring C(X) of all real-valued continuous functions on X.It is not surprising therefore that the development of the paper closely follows the theory of C(X).During the past twenty years extensive work has been done on the ring C(X).The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X.A significant part of this work has recently been summarized in the book [2].Concerning the ring C(X, Z), very little has been written.This is natural, since C(X, Z) is less important in problems of topology and analysis than C(X).Nevertheless, for some problems of topology, analysis and algebra, C(X, Z) is a useful tool.Moreover, a comparison of the theories of C(X) and C(X, Z) should illuminate those aspects of the theory of C(X) which derive from the special properties of the field of real numbers.For these reasons it seems worthwhile to devote some attention to C(X, Z).The paper is divided into six sections.The first of these treats topological questions.An analogue of the Stone-Cech compactification is developed and studied.In §2, the ideals in C(X, Z) are related to the filters in a certain lattice of sets.The correspondence is similar to that which exists between the ideals of C(X) and the filters in the lattice of zero sets of continuous functions on X.This theory provides a characterization of those ideals of C(X, Z) which are intersections of maximal ideals.§3 is concerned with the space of maximal ideals in C(X, Z).In §4, some existence theorems for maximal ideals are proved.The residue class fields of C(X, Z) modulo maximal ideals are studied in the last two sections.It turns out that those of prime characteristic are trivial: the integers modulo the characteristic.The residue class fields of characteristic zero are distinctly nontrivial.In §5, the cardinality of such fields is investigated.The main result is that they are always uncountable.In §6, the algebraic properties of the zero characteristic residue class fields are examined.It is shown for example that these fields are always quasialgebraically closed.As we noted above, very little has been published concerning the ring C(X, Z).Nevertheless, a considerable number of "folk theorems" exist in the subject.One of our objectives in writing this paper is to get these results

Introduction.The purpose of this paper is to study the ring C(X, Z) of all integer-valued continuous functions on a topological space X.Our subject is similar in many ways to the ring C(X) of all real-valued continuous functions on X.It is not surprising therefore that the development of the paper closely follows the theory of C(X).During the past twenty years extensive work has been done on the ring C(X).The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X.A significant part of this work has recently been summarized in the book [2].Concerning the ring C(X, Z), very little has been written.This is natural, since C(X, Z) is less important in problems of topology and analysis than C(X).Nevertheless, for some problems of topology, analysis and algebra, C(X, Z) is a useful tool.Moreover, a comparison of the theories of C(X) and C(X, Z) should illuminate those aspects of the theory of C(X) which derive from the special properties of the field of real numbers.For these reasons it seems worthwhile to devote some attention to C(X, Z).The paper is divided into six sections.The first of these treats topological questions.An analogue of the Stone-Cech compactification is developed and studied.In 2, the ideals in C(X, Z) are related to the filters in a certain lattice of sets.The correspondence is similar to that which exists between the ideals of C(X) and the filters in the lattice of zero sets of continuous functions on X.This theory provides a characterization of those ideals of C(X, Z) which are intersections of maximal ideals.3 is concerned with the space of maximal ideals in C(X, Z).In 4, some existence theorems for maximal ideals are proved.The residue class fields of C(X, Z) modulo maximal ideals are studied in the last two sections.It turns out that those of prime characteristic are trivial: the integers modulo the characteristic.The residue class fields of characteristic zero are distinctly nontrivial.In 5, the cardinality of such fields is investigated.The main result is that they are always uncountable.In 6, the algebraic properties of the zero characteristic residue class fields are examined.It is shown for example that these fields are always quasialgebraically closed.As we noted above, very little has been published concerning the ring C(X, Z).Nevertheless, a considerable number of "folk theorems" exist in the subject.One of our objectives in writing this paper is to get these results

Let PQC stand for the set of all piecewise quasicontinuous functions on the unit circle, i.e., the smallest closed subalgebra of \( L^{\infty}\,(\mathbb{T})\) which contains the classes of all piecewise continuous functions PC and all quasicontinuous functions \( QC \, = \, (C\,+\,H^{\infty})\,\cap\,(C\,+\,\overline{H^\infty})\). We analyze the fibers of the maximal ideal spaces M(PQC) and M(QC) over maximal ideals from \(M(\widetilde{QC})\) where \(\widetilde{QC}\) stands for the C* algebra of all even quasicontinuous functions. The maximal ideal space \(M(\widetilde{QC})\) is described and partitioned into various subsets corresponding to different descriptions of the fibers.

Let G be a locally compact abelian group with dual group F. Let M(G) be the algebra of bounded regular Borel measures on G under convolution multiplication, and let L(G) be the subalgebra of M(G) consisting of all absolutely continuous measures in M(G). By an L-subalgebra of M(G) we mean a closed subalgebra N, with the property that if k E N and v E M(G), with v absolutely continuous with respect to pu, then v E N. Clearly L(G) is an L-subalgebra of M(G). Let N be an L-subalgebra of M(G). If y E r and h,(1k) = f S d[k for Zt E N, then hy is a multiplicative linear functional on N. Hence, if A is the space of all multiplicative linear functionals on N, then y -h, maps r into A. If this map is one to one and onto, then we shall say that the maximal ideal space of N is F. Note that the maximal ideal space of L(G) is r (cf. [5, Chapter 1]), but L(G) is not unique in this respect. If we define (L(G))112 to be the intersection of all maximal ideals of M(G) containing L(G), then (L(G))112 is also an L-subalgebra of M(G) with maximal ideal space F (cf. [7, Lemma 1 and Theorem 1]). If e E M(G) and Aun E L(G) for some n, then tu E (L(G))112. Hewitt and Zuckerman have recently shown in [3], that for every nondiscrete l.c.a. group G, there is a singular measure Z E M(G) such that 2 E L(G). This shows that L(G) = (L(G))112 if G is nondiscrete. Our main theorem (Theorem 1) characterizes completely those L-subalgebras of M(G) that have maximal ideal space r. This result was conjectured in [7] and proved there for a special case. The missing ingredient for a proof in the general case is supplied by one of the results of [8].

This paper is concerned with generalizations to F-algebras of theorems which Gleason has proved for finitely generated maximal ideals in Banach algebras. Let A be a uniform commutative F-algebra with identity such that Spec (A) is locally compact; let x be a nonisolated point of Spec (A), and let ker(x) denote the maximal ideal of all elements of A which vanish at x. In this paper it is shown that: If f is an element of A vanishing only at x, then the principal ideal Af generated by f is closed in A. If the polynomials in the elementf are dense in A and if ker (x) is finitely generated, then there exists an open set U containing x such that ker(y) is generated by f f(y) for all y in U. An example is given which shows that if A is not uniform, the conclusion of the last result may not be true. In fact, the example shows that it is possible to have a nonisolated finitely generated maximal ideal in the algebra. A second example shows that in a uniform Falgebra with locally compact spectrum, ker(x) can be generated by an elementf such thatf f(y) generates no other ker(y) even when the ker(y) are principal. Introduction. The results in this paper generalize to F-algebras results which are known for Banach algebras (see Theorem 2.1(ii) and Theorem 2.2 of [4]). Although the results stated are only for principal maximal ideals, they should point out some of the difficulties in general for finitely generated maximal ideals. Suppose that B is a commutative Banach algebra with identity. Gleason [4] proved the following theorems dealing with the generators of an algebraically finitely generated maximal ideal. (G 1) If I is a maximal ideal of B which is generated by gl, ..., gn,X then there exists a neighborhood U of I in the maximal ideal space of B such that each maximal ideal M in U is generated by g1 ^ (M), ..., gn g (M). (G2) If the subalgebra generated by 1, Z1, . . , Zk is dense in B and if I is a finitely generated maximal ideal in B, then I is generated by z1 Zj (I), Zk -Zk (I ). One immediate consequence of these theorems is that in a commutative Banach algebra with identity, the set of finitely generated maximal ideals is an open set. We will see with an example that this need not be true for F-algebras. Furthermore, we will show that when one considers F-algebras the conclusion Presented to the Society, January 25, 1973; received by the editors February 18, 1976. AMS (MOS) subject classifications (1970). Primary 46H99; Secondary 46E25.

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms