The ring of entire functions
Jonas Bjermo
U.U.D.M. Project Report 2004:15
Examensarbete i matematik, 20 poäng Handledare och examinator: Karl-Heinz Fieseler
Juni 2004
Department of Mathematics
The ring of entire functions.
Examensarbete
Matematiska Institutionen Uppsala Universitet
Jonas Bjermo August 19, 2004
Contents
1 Introduction 2
2 Holomorphic and meromorphic functions 2
2.1 Holomorphic functions 2
2.2 Meromorphic functions 5
3 Preparation for the ideal theory in O(G) 8
3.1 Ideals 8
3.2 Convergent series and sequences of complex functions 9
3.3 Infinite products of complex functions 11
3.4 Mittag-Leffler series 13
3.5 Greatest common divisors 15
3.6 Wedderburn’s lemma 16
4 Ideal structure in O(G) 17
4.1 Principal ideal theorem 17
4.2 Fixed and free ideals 19
4.3 Maximal ideals 21
5 Prime ideals 26
6 References 29
Acknowledgment
I am most grateful to my supervisor Karl-Heinz Fieseler for presenting the idea of this thesis and for answering all my questions about this thesis and related subjects.
1 Introduction
In 1940, O. Helmer [5] proved that, in the ring of all entire functions, every finitely generated ideal is a principal ideal. In this text we show, based on [6, chapter 6.3], that this is also valid in rings of functions holomorphic in arbi- trary open connected subsets G ⊂ C denoted O(G). To prove this, several tools from complex function theory are used, such as the concept of greatest common divisors which depends on Weierstraß products, and Mittag-Leffler series which help us to prove the important lemma of Wedderburn.
Then, as in [3] and [4], we consider the ideals in the ring O(G) with the difference that we use the concept of divisors to describe them. We introduce so called filters and deal with them instead of the distributions of zeros for the functions in the ideal. Both maximal and prime ideals are considered. We will see that the factor ring O(G)/m, where m is maximal is isomorphic to C both when the functions in m have at least one common zero and when they have no common zeros.
2 Holomorphic and meromorphic functions
2.1 Holomorphic functions
We will start to recall some results from complex analysis. A domain is a non-empty, open subset of C. From now on it will always be denoted D. A connected domain is called a region and will be denoted G in the rest of the text.
Definition A function f : D → C is called holomorphic in the domain D if f is complex-differentiable at every point of D; f is called holomorphic at c ∈ D if there is an open neighborhood U of c lying in D such that the restriction f |U of f to U is complex differentiable in U .
Remark The set of all points at which a function is holomorphic, is al- ways open in C. A function which is holomorphic at c is always complex- differentiable at c, but the converse do not need to be true.
The set of all function holomorphic in D is denoted O(D). Let us recall from complex analysis that for a power series P∞n=0an(z − c)n there exists a unique number R ≥ 0 (possibly R = +∞), such that if |z − c| < R, the series converges, and if |z − c| > R, the series diverges. The convergence is uniform and absolute on every closed disc in A = {z ∈ C: |z − c| < R}.
R is called its circle of convergence. We also know that a power series P∞
n=0an(z − c)n is holomorphic inside its radius of convergence. On the other hand we recall that a function holomorphic on a domain D equals the Taylor series P∞n=0f(n)n!(c)(z − c)n on every open disc U centered at c and contained in D. That is, f is holomorphic on a domain D if and only if f equals a convergent power series around c on some open disc Br(c) ∈ D.
Definition Let f ∈ O(D), then the set Z(f ) = {z ∈ C: f (z) = 0} is called its zero-set.
Definition If M is any subset of O(G), let Z(M ) = {Z(f )|f ∈ M }.
The zero-set Z(f ) of a function f 6= 0 holomorphic in a region G is a discrete and relatively closed (and hence finite or countably infinite) subset of G [7 p. 232].
Now we define the order of a zero of a holomorphic function.
Definition Let f ∈ O(D) and let c ∈ D, then f has a zero of order m at c if f (c) = 0, ..., fm−1(c) = 0, fm(c) 6= 0.
The function f being holomorphic means that we can expand it in a Taylor series f (z) = P∞n=0f(n)n!(c)(z − c)n. Then we see that f has a zero of order m if and only if in a neighborhood of c, we can write f (z) = (z − c)mh(z) where h is holomorphic at c, h(z) = P∞µ≥mf(µ)µ!(c)(z − c)µ−m and h(c) = f(m)(c)/m! 6= 0.
The order is denoted oc(f ) and we set oc(f ) := ∞ if f equals zero near c.
We have a rule for computation of the order.
Theorem 2.1 (Product rule) If the functions f and g are holomorphic near c, then oc(f g) = oc(f ) + oc(g).
Proof Let oc(f ) = n and oc(g) = m. Using the fact that n + ∞ = ∞, the definition of the order and Taylors theorem, we get that oc(f g) = m + n = oc(f ) + oc(g).
It is easy to see that the set O(D) becomes a commutative ring with unity with the pointwise addition and multiplication. The units in O(D) are exactly the nowhere vanishing functions. We will now see that if D is connected, which means that D is a region in C then the ring O(D) is an integral domain.
Theorem 2.2 (Identity Theorem [7 p. 228]) The following statements about a pair f , g of holomorphic functions in a region G ⊂ C are equivalent:
1. f = g.
2. The coincidence set {w ∈ G|f (w) = g(w)} has a cluster point in G.
3. There is a point c ∈ G such that f(n)(c) = g(n)(c) for all n ∈ N.
Proof 1) ⇒ 2) is trivial.
2) ⇒ 3) Set h := f − g ∈ O(G). Then by the hypothesis the zero- set M := {w ∈ G|h(w) = 0} has a cluster point c ∈ G. If there is an m ∈ N with h(m)(c) 6= 0 then we consider the smallest such m. From above we see that we have the factorization h(z) = (z − c)mhm(z) with hm(z) = Pµ≥mh(µ)µ!(c)(z − c)µ−m ∈ O(B) for all open balls B centered at c in G and hm(c) 6= 0. Because of its continuity hm is zero free in some neighborhood U ⊂ B of c. Then M ∩ (U \ {c}) = ∅, that is c is not a cluster point of M . This contradiction shows that there is no such m, that is h(n)(c) = 0 for all n ∈ N, hence f(n)= g(n) for all n ∈ N.
3) ⇒ 1) Set h := f − g, and let Sk= Z(h(k). Each Skis relatively closed in G because h(k) ∈ O(G) is continuous. Then S := ∩∞0 Sk is relatively closed in G. However, if z1 ∈ S then the Taylor series of h in any open B ⊂ C centered at z1 equals the zero series. Hence h(k)|B = 0 for every k ∈ N, that is B ⊂ S. This implies that S is open in G. Since G is a region (hence connected), G is the only non-empty subset of G which is both open and closed. Hence S = G, and therefore f = g.
A consequence of the Identity Theorem is
Theorem 2.3 The domain D ⊂ C is connected if and only if the ring O(D) is an integral domain.
Proof ⇒) Suppose f , g ∈ O(D), f is not the zero function but f g is, that is f (z)g(z) = 0 for all z ∈ D. There is some c ∈ D where f (c) 6= 0 and a neighborhood U ⊂ D of c in which f is zero-free. Then g(U ) = {0} and since D is a region (hence connected) this means that g(D) = {0} by the Identity Theorem. That is, g = 0, the zero element of the ring O(D).
⇐) If D is not a region then it would not be connected. It would be possible to express D as the disjoint union of the non-empty open sets D1 and D2. Let us define f , g in D by
f (z) := 0 for z ∈ D1 and f (z) := 1 for z ∈ D2, g(z) := 1 for z ∈ D1 and g(z) := 0 for z ∈ D2.
These functions are holomorphic in D, and neither is the zero function in O(D) but f g is. This contradicts the hypothesis that O(D) is an integral domain.
2.2 Meromorphic functions
The Taylor expansion does not apply to functions that fail to be holomorphic at some points. For such functions we have another expansion called the Laurent expansion. Let r1 ≥ 0, r2> r1, and c ∈ C, and consider the region A = {z ∈ C: r1< |z − c| < r2}. Let us recall from complex analysis that if a function f is holomorphic in A, it can be written f (z) =P∞n=0an(z − c)n+ P∞
n=1 bn
(z−c)n where both series on the right side of the equation converge absolutely on A and uniformly on any set of the form Bρ1,ρ2 = {z ∈ C: ρ1 ≤
|z − c| ≤ ρ2} where r1 < ρ1 < ρ2 < r2. The series is called the Laurent series or the Laurent expansion around c in A. The seriesP∞n=0an(z − c)n, respectively,P∞n=1(z−c)bn n is called its regular, respectively, principal part.
When r1 = 0 in the Laurent expansion we have a special case. In this case, f is holomorphic in {z : 0 < |z − c| < r2}, and we say that c is an isolated singularity of f .
Definition Let f be holomorphic in a domain D except for one point c ∈ D, i.e., f is holomorphic in D \ {c}, then c is called an isolated singularity of f.
As seen above, if f is holomorphic in {z : 0 < |z − c| < r2}, then we can expand f in a Laurent series: f (z) = ... +(z−c)bn n+ ... +(z−c)b1 + ao+ a1(z − c) + a2(z − c)2+ ... valid for 0 < |z − c| < r2. There are three different types of isolated singularities depending on the number of coefficients bn 6= 0. If all the coefficients bn6= 0 is zero then we say that c is a removable singularity, and if the number of bn are infinite c is called an essential singularity. We are interested in the case when all but a finite number of the coefficients bn is zero.
Definition If c is an isolated singularity of f and if all but a finite number of the bn in the Laurent expansion are zero, then c is called a pole of f. The highest integer k such that bk6= 0, is called the order of the pole.
Definition A meromorphic function on D is a pair (f, P ), where P ⊂ D is discrete and f : D \ P → C is holomorphic with a pole at each point of P . The set P is called the pole-set of f . Usually we write f instead of (f, P ) and P = P (f ).
The set of all function meromorphic in a domain D is denoted M(D).
Remark Because the pole-set is allowed to be empty, the functions holo- morphic in D are also meromorphic in D.
The pole-set of each function f 6= 0 meromorphic in D is, just as for the zero-set of a holomorphic function, a discrete and relatively closed subset of
D. Then it follows that the pole-set of each function meromorphic in D is either empty, finite, or countably infinite [7 p. 316].
From complex analysis it is known that poles (locally) arise via the formation of reciprocals of holomorphic functions.
Proposition 2.4 Let f ∈ O(D\{c}) and let c ∈ D be an isolated singularity of f , then c is a pole of order m if and only if there exists a function g ∈ O(D), g(c) 6= 0, such that f (z) = (z−c)g(z)m for z ∈ D \ {c}.
Proof ⇒) By definition c is a pole iff f (z) = (z−c)bk k + (z−c)bk−1k−1 + ... +
b1
(z−c) +P∞n=0an(z − c)n = (z−c)1 k(bk+ bk−1(z − c) + ... + b1(z − c)k−1 + P∞
n=0an(z − c)n+k), where bk 6= 0. This is valid in D \ {c}. Let g(z) = bk+ bk−1(z − c) + ... + b1(z − c)k−1+P∞n=0an(z − c)n+k, this implies that g ∈ O(D) and g(c) = bk6= 0.
⇐) Let g ∈ O(D) where g(z) = P∞n=0an(z − c)n and set f = (z−c)g(z)n. Then f =P∞n=0an(z − c)n−m, and by definition, it has a pole at c of order m ≥ 1.
Theorem 2.5 Let f ∈ O(D \ {c}), then f has a zero of order m at c if and only if f (z)1 has a pole of order m at c. If ˆf ∈ O(D) and ˆf (c) 6= 0 then f (z)f (z)ˆ also has a pole of order m at c.
Proof The function f ∈ O(D \ {c}) has a zero of order m iff f (z) = (z − c)mh(z), where h(z) is holomorphic at c with h(c) 6= 0. Set g(z) := h(z)1 , g ∈ O(D), g 6= 0. Then f (z) = (z − c)mg(z)1 , by prop 2.4 this is equivalent to that f (z)1 has a pole of order m at c.
As we already know, a function f with a pole of order m at c can be written f (z) =P1n=−mbn(z −c)n+P∞n=0an(z −c)n. We can then see that to ask when f has a pole of order m at c is the same as asking when (z − c)mf is bounded.
Theorem 2.6 Let m ∈ N, m ≥ 1 and let f ∈ O(D \ {c}). Then f has a pole of order m at c if and only if there is a neighborhood U of c lying in D and positive finite constants M∗, M∗ such that for all z ∈ U \ {c}
M∗|z − c|−m ≤ |f (z)| ≤ M∗|z − c|−m.
Proof ⇒) The function f has a pole of order m at c iff h := f1 has a zero of order m at c, which can be written in the form (z − c)mh for anˆ ˆh ∈ O(U ), U ⊂ D small enough. Let M∗ = infz∈U{|ˆh(z)|−1} > 0 and M∗ = supz∈U{|ˆh(z)|−1} < ∞. The claim now follows from |f (z)| = |z − c|−m|ˆh(z)|−1.
⇐) |(z − c)mf (z)| ≤ M∗ for z ∈ U \ {c} shows that (z − c)mf is bounded near c and |(z − c)m−1f (z)| ≥ M∗|z − c|−1 shows that (z − c)m−1f is not bounded near c. That is, c is a pole of order m.
Let f ∈ O(D). We say that f increases uniformly to ∞ around c, written limz→cf (z) = ∞, if for every finite M there is a neighborhood U of c in D such that infz∈U \{c}|f (z)| ≥ M
Then a consequence of theorem 2.6 is the following
Corollary 2.7 The function f ∈ O(D \ {c}) has a pole at c if and only if limz→cf (z) = ∞.
In view of this corollary, at each pole, we choose f (z) := ∞ for z ∈ P (f ).
So meromorphic functions are special mappings from D ⊂ C to C ∪ {∞}.
If f , g ∈ M(D) with pole-sets P (f ), P (g) are given, then P (f ) ∪ P (g) is also discrete and relatively closed in G. In G \ P (f ) ∪ P (g) both f and g are holomorphic, and hence f +g, f g are holomorphic. For each c ∈ P (f )∪P (g), f and g can be written as Laurent expansions with finite principal part in an open neighborhood U of c, with U ∩ (P (f ) ∪ P (g)) = {c}. Then both f + g and f g can be written as Laurent expansions with finite principal part. The point c is then either a pole or a removable singularity of f + g and f g. Thus the pole-set of these functions are subsets of P (f ) ∪ P (g), which are discrete and relatively closed in G. This implies that f + g, f g ∈ M(D). From the rules of calculating with holomorphic functions it follows that M(D) is a commutative ring with unity with respect to pointwise addition and multiplication. The ring O(D) is a subring of M(D).
In the ring O(D), division of an element f is possible only when the value of f is zero-free in D. But in the ring M(D), division of functions which have zeros in D is possible.
Now we define the zero-set of a meromorphic function.
Definition Let f ∈ M(D), then Z(f ) is called the zero-set of f if Z(f ) is the zero-set of the holomorphic function f |(D \ P (f )) ∈ O(D \ P (f )).
Remark Z(f ) is relatively closed in G and Z(f ) ∩ P (f ) = ∅.
Now we define the order function for a meromorphic function. If a func- tion f 6= 0 is meromorphic, then it can be developed uniquely into a Laurent series P∞m aν(z − c)ν with aν ∈ C, m ∈ Z and am 6= 0.
Definition If f 6= 0, f ∈ M(D) the number m in the Laurent series above is called the order of f at c and is denoted oc(f ).
From the definition it follows directly that for an f meromorphic at c:
1. f holomorphic at c ⇔ oc(f ) ≥ 0
2. If m = oc(f ) < 0, then c is a pole of f of order −m.
As for holomorphic functions the product rule 2.1 is also valid for mero- morphic functions.
Now we prove that the ring of all functions meromorphic in a region is a field.
Theorem 2.8 Let u ∈ M(D), then u is invertible, that is uv = 1 for some v ∈ M(D), if and only if the zero-set Z(u) is discrete in D.
Proof ⇒) For c ∈ D, uv = 1 implies that u(c) = 0 iff v(c) = ∞ and u(c) = ∞ iff v(c) = 0. This means that Z(u) = P (v) and P (u) = Z(v), then by the definition of a meromorphic function the zero-set Z(u) is discrete.
⇐) The set Z(u) ∪ P (u) is discrete and relatively closed in G. Choose v := 1/u, then v is holomorphic in G \ Z(u) ∪ P (u) and have a pole at every point of Z(u). Every point c ∈ P (u) is a removable singularity of v because limz→c1/u(z) = 0. Hence v ∈ M(D)
From this theorem we see that the quotient of two elements f, g ∈ M(D) exists in the ring M(D) if Z(g) is discrete in D.
A consequence of theorem 2.8 is
Corollary 2.9 Let G ⊂ C be a region, then M(G) is a field.
Proof If f ∈ M(G), f 6= 0 and G is a region, then G \ P (f ) is a region and f |(G \ P (f )) is a holomorphic function which is not the zero element of O(G). Therefore Z(f ) is discrete in G and then f is invertible. Hence, every element of M(G) \ {0} is invertible and therefore M(G) is a field.
Because M(G) is a field it contains no proper ideals 6= {0}. The ring O(G) on the other hand, has an interesting ideal structure.
3 Preparation for the ideal theory in O(G)
To prove some important result about the ideal structure in the integral domain O(G) we make use of some important tools. These are the lemma of Wedderburn and the concept of greatest common divisors. In the proof of Wedderburns lemma we make use of Mittag-Leffler series which will also be considered. We start with ideals.
3.1 Ideals
Since we are going to consider the ideal structure in the ring of functions holomorphic in open connected subsets of C we give some basic definitions.
In this subsection R is a commutative ring with unity.
Definition Let R be a ring. A subset I ⊂ R, I 6= ∅ is called an ideal in R if I is an additive subgroup and ax ∈ I for all a ∈ I, x ∈ R.
Now, let M 6= ∅ be any subset of R and L be the set of all finite linear combinations i.e. L = {u =Pni=1rifi : ri ∈ R, fi ∈ M }. Then L is an ideal in R and M is said to generate L.
Definition An ideal I in a ring R is called finitely generated if there is a finite set that generates I. A ring R is called Noetherian if every ideal in R is finitely generated.
If we can choose M with only one element, we have a special case.
Definition An ideal I in a ring R is called principal if it is generated by an element f , i.e. if I = {rf : r ∈ R} for some f ∈ R. A integral domain R is called a principal ideal domain if every ideal is a principal ideal.
Remark A finitely generated ideal generated by M = {f1, ..., fr} is usually denoted I = Rf1+ ... + Rfn, and a principal ideal generated by f is usually denoted (f ) or Rf .
Definition An ideal I in a ring R is called maximal provided that, I 6= R and there does not exist an ideal J 6= R such that I ⊂ J.
Definition An ideal I 6= R is called a prime ideal if ab ∈ I implies that either a ∈ I or b ∈ I for a, b ∈ R.
3.2 Convergent series and sequences of complex functions Let X be a metric space.
Definition A sequence of functions fn: X → C is called uniformly conver- gent in A ⊂ X to f : A → C if for every > 0 there exists an n ∈ N such that |fn(x) − f (x)| < for all n ≥ n and all x ∈ A.
A seriesPfν of functions converges uniformly in A if the sequence sn= Pnfν of partial sums converges uniformly in A.
Remark The limit functions f andPfν are uniquely determined.
We introduce the supremum semi-norm |f |A:= sup{|f (x)|: x ∈ A} ≤ ∞ for subsets A ⊂ X and functions f : X → C. The set V := {f : X → C: |f |A< ∞} is a C-vector space and the mapping f → |f |A fulfills:
1. |f |A= 0 ⇔ f |A= 0.
2. |cf |A= |c||f |A.
3. |f + g|A≤ |f |A+ |g|A. for all f, g ∈ V , c ∈ C.
We see that the sequence fnconverges uniformly in A to f exactly when lim |fn− f |A= 0.
Since convergence does not need to be uniform on the whole X, we introduce the following
Definition A sequence of functions fn: X → C is called locally uniformly convergent in X if every point x ∈ X lies in a neighborhood Ux in which the sequence fn converges uniformly.
A series Pfν is called locally uniformly convergent in C when its asso- ciated sequence of partial sums is locally uniformly convergent in X.
Uniform convergence implies locally uniform convergence. Let A1, ...Am be subsets of X. If the sequence fn: X → C converges uniformly in each of these subsets, then it also converges uniformly in the union A1∪ ... ∪ Am. A consequence of this is
Theorem 3.1 If the sequence fn converges locally uniformly in X, then it converges uniformly on each compact subset K of X.
Proof Every point x ∈ K has an open neighborhood Ux in which fn is uniformly convergent. The open cover {Ux: x ∈ K} of the compact set K admits a finite subcover, say Ux1, ..., Uxm. Then fn converges uniformly in Ux1 ∪ ... ∪ Uxm, and therefore so in the subset K of this union.
Definition A sequence or series converges compactly in X if it converges uniformly on every compact subset of X.
So we may reformulate 3.1:
Corollary 3.2 Local uniform convergence implies compact convergence.
Let X be a metric space. We call X locally compact if each of its points has at least one compact neighborhood. Then it is easy to see the following Theorem 3.3 If X is locally compact, then every compactly convergent se- quence or series in X is locally uniformly convergent in X.
In locally compact spaces local uniform convergence and compact con- vergence are equivalent. This is also valid in domains in C since they are locally compact.
To guarantee that every re-arrangement of a series will converge locally uniformly we introduce the following concept.
Definition A seriesPfν of functions fν: X → C is called normally conver- gent in X if each point of x has a neighborhood U which satisfiesP|fν|U <
∞.
By the Weierstraß majorant criterion [7 p.103], we see that every series which is normally convergent in X is locally uniformly convergent in X.
Since locally uniform convergence implies compact convergence we get that:
IfPfν is normally convergent in X, thenP|fν|K < ∞ for every compact subset K ⊂ X.
If the metric space is locally compact then locally uniform convergence and compact convergence is equivalent, hence the converse is true:
If X is locally compact and P|fν|K < ∞ for every compact subset K ⊂ X, thenPfν is normally convergent in X.
Since domains D in C are locally compact then Pfν is normally con- vergent in D if and only ifP|fν|K < ∞ for every compact set K ⊂ D.
We end this section with the rearrangement theorem.
Theorem 3.4 If P∞0 fν converges normally in X to f , then for every bi- jection τ : N → N the rearranged seriesP∞0 fτ (ν) also converges normally in X to f .
Proof For every point x ∈ X there is a neighborhood U for whichP|fν|U <
∞. Since for every bijection τ of N every rearrangement of a absolutely convergent series of complex numbers converges to the same limit as the original one (see [7 p.28]), we haveP|fτ (ν)|U < ∞ andP∞0 fτ (ν)(x) = f (x).
This is valid for every x ∈ X, hencePfτ (ν) converges normally in X to f .
3.3 Infinite products of complex functions
Definition Let (aν)ν≥k be a sequence of complex numbers. The sequence (Qnν=k)n≥kaν of partial products is called an infinite product with the factors aν. It is writtenQ∞ν=kaν orQaν. In general, k = 0 or k = 1.
We introduce the partial products pm,n := amam+1...an =Qnν=maν for k ≤ m ≤ n.
Definition The product Qaν is called convergent if there exists an index m such that the sequence (pm,n)n≥m has a limit ˆam6= 0.
We call a := akak+1...am−1ˆam the value of the product and we write Qaν := akak+1...am−1aˆm = a. The following result is easy to see, namely:
That a productQaν being convergent is equivalent to the fact that at most finitely many factors are zero and the sequence of partial products consisting of the nonzero elements has a limit 6= 0. We also see that a convergent productQaν is zero if and only if at least one factor is zero.
Theorem 3.5 If Q∞ν=0aν converges, then ˆan :=Q∞ν=naν exists for all n ∈ N. Moreover, lim ˆan= 1 and lim an= 1
Proof We assume that a := Qaν 6= 0. Then ˆan = a/p0,n−1. Since lim p0,n−1 = a, it follows that lim ˆan = 1. And lim an = 1 holds because ˆ
an6= 0 and an= ˆan/ˆan+1.
Now we let X denote a locally compact metric space.
Definition Let fν ∈ C(X) be a sequence of continuous functions on X with values in C. The infinite product Qfν is called compactly convergent in X if, for every compact set K ⊂ X there exists an index m such that the sequence pm,n := fmfm+1...fn, n ≥ m, converges uniformly on K to a non-vanishing function ˆfm.
For each point x ∈ X f (x) :=Qfν(x) ∈ C exists and we call the function f : X → C the limit of the product and write f =Qfν.
The continuity theorem [7 p. 95], says that if the sequence fν ∈ C(X) converges locally uniformly in X, then the limit function f = lim fn is likewise continuous on X. Because locally uniform convergence and compact convergence coincide in the locally compact metric space X, it follows from the continuity theorem and theorem 3.5 that
if Qfν converges compactly to f in X, then f is continuous in X and the sequence fν converges compactly in X to 1.
In view of this we write the factors of a product Qfν in the form fν = 1 + gν and the sequence gν converges compactly to zero if Qfν converges compactly.
Definition A product Qfν with fν = 1 + gν ∈ C(X) is called normally convergent in X if the seriesPgν converges normally in X.
It is quite easy to see that [6 p. 7] if Qν≥0fν converges normally in X, then for every bijection τ : N → N, the product Qν≥0fτ (ν) converges nor- mally in X and the product converges compactly in X. It actually converges compactly to a function f : X → C, see the rearrangement theorem [6 p. 8].
We will need the next theorem (3.7) in section 4.2. First we recall from complex analysis that if a sequence fn of functions holomorphic in D con- verges compactly there to f , then f is holomorphic in D. This is known as Weierstraß convergence theorem. This theorem can be sharpened as follows:
Let G be a bounded region and fn a sequence of functions which are continuous on G and holomorphic in G. If the sequence fn|∂G converges uniformly on ∂G, then the sequence fn converges uniformly in G to a limit function which is continuous on G and holomorphic in G (see [7 p. 269] for proof).
From this we get the following
Lemma 3.6 If A is a discrete subset of G and fn ∈ O(G) is a sequence which converges compactly in G\A, then the sequence fnconverges compactly in the whole of G.
Theorem 3.7 If f =Qfν, fν ∈ O(G), is normally convergent in G, then the sequence ˆfn=Qν≥nfν ∈ O(G) converges compactly in G to 1.
Proof Let ˆfm6= 0. Then A := Z( ˆfm) is discrete and relatively closed in G.
All the partial products pm,n−1∈ O(G), n > m, are non-vanishing in G \ A and
fˆn(z) = ˆfm(z)(p 1
m,n−1(z)) for all z ∈ G \ A.
The sequence 1/pm,n−1converges compactly in G \ A to 1/ ˆfm. Therefore by lemma 3.6, this sequence also converges compactly in G to 1.
3.4 Mittag-Leffler series
We know that in every Laurent expansion, P∞1 bµ(z − c)−µ was called the principal part. It is called finite if almost all bµ vanish. A distribution of principal parts on D is defined to be a function ϕ that maps every point d ∈ D ⊂ C to a principal part qd, such that the set T = {z ∈ D|ϕ(z) 6= 0}, is discrete. This set is called the support of ϕ.
Every function h holomorphic in D except for some isolated singularities can be written as a Laurent series with a principal part around its singu- larities, every such function determines thus a principal part distribution P D(h) where the support is the set of nonremovable singularities.
If all except a finite number of the bn vanish in all the principal parts of the Laurent expansions for some function h holomorphic in D \ {d1, d2, ...}, the points {d1, d2, ...} are poles. Then h is a meromorphic function in D.
Thus h is meromorphic in D if and only if P D(h) is a distribution of finite principal parts with the pole-set P (h) as support. We now pose the following problem:
For every principal part distribution ϕ with support T , construct a func- tion h ∈ O(D \ T ) with P D(h) = ϕ.
We are able to construct such functions with help of certain series called Mittag-Leffler series for a principal part distribution ϕ, and show that every such series represents a meromorphic function with principal part distribu- tion equal to ϕ. Let us define Mittag-Leffler series.
We let T = {d1, d2, ...} be the support of the principal part distribution ϕ and arrange the points d1, d2, ... in a sequence where every point occurs once. If the origin belongs to T we set d1 = 0. The principal part is uniquely described by the sequence (dν, qν) where qν := ϕ(dν).
Definition A series h =P∞ν=1(qν− gν) is called a Mittag-Leffler series for the principal part distribution (dν, qν) on D if the following holds
1. gν is holomorphic in D;.
2. the series h converges normally in D \ {d1, d2, ...}.
Proposition 3.8 If h is a Mittag-Leffler series for a principal part distri- bution (dν, qν), then h ∈ O(D \ {d1, d2, ...}) and P D(h) = (dν, qν).
Proof Because the series converges normally in D \ {d1, d2, ...}, h is holo- morphic in D \ {d1, d2, ...}. Since all the summands qν − gν, ν 6= n, are holomorphic in a neighborhood U ⊂ D of dn, the series Pν6=n(qν− gν) con- verges uniformly on every compact disc in U to a function fn∈ O(U ) in U , by Weierstraß convergence theorem [2 p. 213]. Since h = fn+ (qn− gn) we have h − qn = fn − gn in U \ {dn} where f and gn are holomorphic at dn, it follows that qn is the principal part of h at dn, n ≥ 1. Hence P D(h) = (dν, qν).
Remark If the principal parts are finite, then the points dν are poles. That means that in the Mittag-Leffler series for the distribution of that finite principal parts, every summand is meromorphic in D. Which implies that the series is a normally convergent series of meromorphic functions in D.
The terms gν inP∞ν=1(qν− gν) which force the series to converge are called the convergence producing summand of the Mittag-Leffler series.
We now consider the case D = C. Every function qν ∈ O(C \ {dν}) can be expanded to a Taylor series around 0, which converges in the disc of radius |dν|, ν ≥ 2. We denote by pνk the kth Taylor polynomial for qν around 0. We show that these polynomials force the series to converge.
Theorem 3.9 (Mittag-Leffler) For every principal part distribution (dν, qν)ν≥1 in C, there exists a Mittag-Leffler series in C of the form q1+P∞ν=2(qν − pνkν), where pνkν: = kνth Taylor polynomial of qν around 0.
Proof Since the sequence (pνk)k≥1 converges compactly in B|dν| to qν, for every ν ≥ 2 we can choose a kν ∈ N such that |qν(z) − pνkν(z)| ≤ 2−ν for all z with |z| ≤ 12|dν|. Since lim dν = ∞, there is for every compact set K ⊂ C a n such that K lies in all the discs B1
2|dν|with ν ≥ n. Therefore P
ν≥n|qν− pνkν|K ≤Pν≥n2−ν < ∞,
which means that the series is normally convergent in C \ {d1, d2, ...}.
The polynomials pνkν are holomorphic in C, hence the series is a Mittag- Leffler series in C for (dν, qν)ν≥1.
Now the following corollary is clear.
Corollary 3.10 Every principal part distribution on C with support T is the principal part distribution of a function holomorphic in C \ T .
Corollary 3.11 Every function h that is meromorphic in C can be repre- sented by a seriesPhν that converges normally in C, where each summand hν is rational and has at most one pole in C.
Proof By Mittag-Leffler’s theorem, corresponding to the principal part dis- tribution P D(h) there is a Mittag-Leffler series ˆh ∈ C with polynomials as convergence producing summands. Because the function h is meromorphic, all principal parts are finite. This implies that all the summands of this series are rational functions with exactly one pole in C. The difference h − ˆh is holomorphic in all of C and can therefore be expanded to a Taylor series.
Hence it is normally convergent.
3.5 Greatest common divisors
In our way to prove the principal ideal theorem in section 4 we are going to use the concept of greatest common divisors in the ring O(G), where G ⊂ C is a region. The most important result shown below is that there exists a greatest common divisor for every non-empty subset S in the integral domain O(G).
Definition A map D: G → Z is called a divisor on G if its support |D| :=
{z ∈ G|D(z) 6= 0} is discrete in G.
Let h be meromorphic in a region G with both zero-set Z(h) and pole-set P (h) discrete in G. Then (h): G → Z determines, by z 7→ oz(h), a divisor with support Z(h) ∪ P (h). Such divisor is called a principal divisor. A divisor D is called positive if D(z) ≥ 0 for all z ∈ G. Positive divisors are also called distributions of zeros. We denote the set of all positive divisors D+(G) Holomorphic functions f have positive divisors (f ).
The basic arithmetic concepts are defined as usual.
Definition A function f ∈ O(G) is called a divisor of g ∈ O(G) if g = f · h with h ∈ O(G).
Remark Note that divisor means two different things here.
There is a connection between divisibility for elements f, g 6= 0 and their principal divisors (f ), (g).
Theorem 3.12 Let f , g ∈ O(G) \ {0}. Then f divides g if and only if (f ) ≤ (g).
Proof The function f divides g if and only if h := g/f ∈ O(G). This occurs if and only if oz(h) = oz(g) − oz(f ) ≥ 0 for all z ∈ G, i.e. if and only if (f ) ≤ (g).
Definition Let S ⊂ O(G), S 6= ∅, then f ∈ O(G) is called a common divisor of S if f divides every element g of S. Let f be a common divisor of S, then f is called a greatest common divisor of S if every common divisor of S is a divisor of f . It is denoted by f = gcd(S).
Remark Greatest common divisors are unique up to unit factors. Despite this we speak about the greatest common divisor f of a set S and denote it f =gcd(S).
Definition A set S ⊂ O(G), S 6= ∅ is called relatively prime if gcd(S)= 1.
Theorem 3.13 Every function f ∈ O(G) with (f ) = min{(g) : g ∈ S, g 6=
0} is a gcd of S 6= {0}.
Proof Let f ∈ O(G). Then (f ) = min{(g) : g ∈ S, g 6= 0} implies that (f ) ≤ (g), which means that f divides g for all g ∈ S. That is f is a common divisor to S. Choose any other common divisor h of S, then (h) ≤ (g) which implies that (h) ≤ (f ) and h divides f . Hence f is a gcd(S).
Theorem 3.14 Every non-empty subset S in O(G) has a gcd.
Proof Given S 6= ∅, choose f ∈ O(G) such that (f ) = min{(g)|g ∈ S, g 6=
0}. Such a function exists by the Weierstraß product theorem [6 p.92].
Theorem 3.15 The set S ⊂ O(G), S 6= ∅ is relatively prime if and only if the functions in S have no common zeros in G.
Proof ⇒) Suppose gcd(S)= 1 and assume that ∩g∈SZ(g) 6= ∅. Then a function vanishing on a point of ∩g∈SZ(g) 6= ∅ with order equal to one would have order less or equal to every g ∈ S. Hence it would be a common divisor of S and thus divide 1, contradiction.
⇐) Let ∩g∈SZ(g) = ∅ = Z(1). Then 1 has order less or equal to every g ∈ S. Hence 1 =gcd(S).
3.6 Wedderburn’s lemma This lemma will help us later on.
Lemma 3.16 Let u, v ∈ O(G) be relatively prime. Then there are functions a, b ∈ O(G) such that au + bv = 1.
Proof Since u, v are relatively prime, we know from theorem 3.15 that Z(u) ∩ Z(v) = ∅. Then the pole set of the function 1/uv is equal to the disjoint union of the pole sets of 1/u and 1/v, that is P (1/uv) = P (1/u) ∪ P (1/v). Because 1/uv is a meromorphic function, by corollary 3.11 of Mittag Leffler’s theorem, it can be written as a normally convergent series Phν, where each summand has at most one pole. This series can be rearranged to a sum of two series a1 = Phν1 and b1 = Phν2, where the poles of the summands Phν1 belongs to P (1/u) = Z(u) and the poles of the summands in hν2 belongs to P (1/v) = Z(v). The sums represents mero- morphic functions, therefore we have: 1/uv = a1+ b1, where a1, b1∈ M(G).
Now we define a: = vb1 and b: = ua1 both holomorphic in G, then it follows that au + bv = 1.
4 Ideal structure in O(G)
4.1 Principal ideal theorem
We recall from algebra that for example in the ring Z or F [x] when F is a field, every ideal is principal and in the ring F [x1, ..., xn] every ideal is finitely generated. The situation in the ring O(G) is a bit different. We know from the previous section that a greatest common divisor exists for every subset of O(G).
Proposition 4.1 If f ∈ O(G) is a gcd of the finitely many functions f1, ..., fn∈ O(G), then f can be written as a sum f = a1f1+ a2f2+ ... + anfn, where a1, ..., an∈ O(G).
Proof By induction on n. Let f 6= 0. For n = 1 it is true. Let n > 1 and choose a function ˆf : = gcd{f2, ..., fn}. By the induction hypothesis we know that ˆf = ˆa2f2 + ... + ˆanfn where ˆa2, ..., ˆan ∈ O(G). Now, because f = gcd{f1, ˆf } we have 1 = gcd{f1/f, ˆf /f }. Then by Wedderburn’s lemma f1/f and ˆf /f satisfy the equation 1 = (f1/f )a + ( ˆf /f )b with functions a, b ∈ O(G), which implies f = f1a + ˆf b = f1a + (f2aˆ2+ ... + fnˆan)b. Let a1: = a and aν: = ˆaνb, ν ≥ 2. Hence f = a1f1+ ... + anfn.
We are now able to prove our main theorem of this section.
Theorem 4.2 (Principal ideal theorem) An ideal I ⊂ O(G) is finitely gen- erated if and only if it is a principal ideal.
Proof ⇒) We reformulate the claim as follows: If I is generated by f1, ..., fn, then I = O(G)f , where f = gcd{f1, ..., fn}. Every non-zero subset in O(G)
has a gcd, then by proposition 4.1, O(G)f ⊂ I. Since f divides all of the f1, ..., fn, we see that f1, ...fn ∈ O(G)f ; hence I ⊂ O(G)f therefore I = O(G)f , which means that I is a principal ideal.
⇐) This direction is trivial.
Since f divides every f1, ..., fn the functions f1, ..., fn vanish where f does, and since f = a1f1+ ... + anfn the function f vanish at the common points where f1, ..., fnvanish, we have Z(f ) = ∩i=1,...,nZ(fi).
However, the next example shows that there exists ideals in O(G) that are not finitely generated.
Example Let G be a region and let A ⊂ G be infinite, discrete and closed in G. The set a := {f ∈ O(G) : f (a) = 0 for all except a finite number of a ∈ A} is an ideal in O(G). Assume that a is finitely generated, i.e. a = O(G)f1+ ... + O(G)fn. Then there exists a kj ∈ N such that for all k ≥ kj, fj(ak) = 0, j = 1, .., n. Choose k0 :=max{k1, ..., kn}, then for h ∈ a, h = Pn
j=1gjfj we have h(ak) = 0 for all k ≥ k0. For any discrete and closed subset T ⊂ G we can construct a holomorphic function f with zero set T . Let T := A \ {a}, then f (a) 6= 0 and f ∈ a, but f is not a linear combination of f1, ..., fn. Contradiction!
Thus we have proved
Proposition 4.3 No ring O(G) is Noetherian, and therefore never a prin- cipal ideal domain.
We see that every ideal in O(G) either is principal or not finitely gener- ated.
Definition An integral domain R is called a unique factorization domain if the following conditions are valid:
1. Every non-unit element p 6= 0 can be factored into a product of a finite number of irreducibles.
2. If p1, ..., pn and q1, ..., qm are two factorizations of the same element of R into irreducibles, then n = m and the qj can be renumbered so that for pi and qi we have pi = qiui, where ui are units in R.
The functions (z − c), c ∈ G are, up to unit factors, precisely the primes of O(G). Functions f 6= 0 in O(G) with infinitely many zeros in G cannot be written as the product of finitely many primes. From the fact that such functions exists, by the general Weierstraß product theorem, we get
Proposition 4.4 No ring O(G) is a unique factorization domain.
4.2 Fixed and free ideals
An element of O(G) is a unit if and only if it vanishes nowhere, therefore every function in a proper ideal vanishes somewhere.
We now distinguish between two types of ideals.
Definition An ideal I in O(G) is called fixed if ∩f ∈IZ(f ) 6= ∅, otherwise it is called free. A point c ∈ G is called a zero of an ideal I if f (c) = 0 for every f ∈ O(G).
Remark A fixed ideal has at least one zero and a free ideal has no zeros.
From the principal ideal theorem 4.2, we have
Corollary 4.5 Any proper finitely generated ideal is fixed.
Definition An ideal is called closed if it contains the limit function of every sequence fn∈ a that converges compactly in G.
We continue with a lemma.
Lemma 4.6 Let I ∈ O(G) be an ideal, where c ∈ G is not a zero, and let f, g ∈ O(G) where f vanishes only at c. If f g ∈ I, then g ∈ I
Proof Choose h ∈ I with h(c) = 1. Let n := oc(f ). If n ≥ 1 then,
f
z−c· g = z−cf · g(h(z) − (h(z) − 1)) =z−cf · gh −h(z)−1z−c · f g ∈ I because h(z)−1z−c has a removable singularity at c and gh ∈ I and f g ∈ I. Applying this n times gives ((z−c)f n) · g ∈ I. Since (z−c)f n is invertible, we have g ∈ I.
By Weierstraß’ product theorem for arbitrary regions [5 p.93], every f 6= 0 that is holomorphic in an arbitrary region G ⊂ C can be written in the form f = uQν≥1fν where u is a unit in the ring O(G) and Qν≥1fν converges normally in G and fν has exactly one zero cν in G of order 1.
Proposition 4.7 If I is a closed free ideal in O(G), then I = O(G).
Proof Let f ∈ I, f 6= 0 and let f =Qfν be a factorization of f as described above. Then ˆfn := Qν≥nfν ∈ O(G) converges compactly in G to 1 by theorem 3.7. We write ˆfn = fnQν≥1fν = fnfˆn+1. Since ˆf0 = f ∈ I and fn has no zeros in G \ {cν}, it follows by induction and lemma 4.6 that ˆfn∈ I for all n ≥ 0. Since I is closed, it follows that 1 ∈ I and hence I = O(G).
Theorem 4.8 An ideal I ∈ O(G) is a principal ideal if and only if it is closed.
Proof ⇒) Let I = O(G)f , and let gn = anf ∈ O(G) converge compactly to g on G. Then an = gn/f converges compactly on G \ Z(f ), by lemma 3.6, to a function a ∈ O(G). Hence g = af ∈ I, so I is closed.
⇐) By 3.14, I has a gcd f in O(G). Then ˆI := f−1I is a free ideal in O(G). Now, ˆI is closed if I is, therefore by proposition 4.7 ˆI = O(G) which implies that I = O(G)f .
From this and proposition 4.7 we have
Corollary 4.9 A proper free ideal in O(G) is never finitely generated.
However, a fixed ideal is not necessarily finitely generated. Let I be a free ideal in O(G), then (z − c)I with c ∈ G would be a fixed ideal that is not finitely generated.
Proposition 4.10 Let I 6= O(G) be a proper free ideal in O(G), then every function f ∈ I has infinitely many zeros.
Proof Assume f ∈ O(G) has only finitely many zeros. Then there are f1, ..., fr ∈ I with Z(f, f1, ..., fr) = ∅, hence 1 = gcd(f, f1, ..., fr) ∈ I. Con- tradiction!
Because a non-zero polynomial only has a finite number of zeros we get Corollary 4.11 No non-zero polynomial belongs to a proper free ideal.
We now describe the ideals, especially the free ones, with help of divisors.
As in section 3.5 we denote D+(G) the set of positive divisors D ∈ D(G), i.e. such that D(z) ≥ 0 for all z ∈ G and D(z) > 0 for at least one z ∈ G.
Definition A non-empty subset F ⊂ D+(G) is called a filter in D+(G) if it satisfies
D0 ≥ D ∈ F =⇒ D0 ∈ F and D, D0 ∈ F =⇒ min(D, D0) ∈ F Furthermore let |F | := {|D|; D ∈ F }.
Proposition 4.12 There is an inclusion preserving bijection between the set of proper ideals 6= {0} in the ring O(G) and the family of all filters in D+(G), namely O(G) ⊃ a 7→ F := (a) := {(f ); f ∈ a \ {0}}
with inverse
F 7→ a := I(F ) := {f ∈ O(G); f = 0 ∨ (f ) ∈ F }
Proof Every function f in an ideal a determines a divisor (f ) by the order function and a contains all other functions that is divided by f . Therefore (a) fulfills, by theorem 3.12 and prop. 4.1, the conditions for a filter.
On the other hand, given a filter F , the set I(F ) ⊂ O(G) is an ideal, since for f, g ∈ I(F )\{0} we have f +g ∈ I(F ) because of (f +g) ≥ min((f ), (g)) ∈ F (when we assume f + g 6= 0) while ”f ∈ O(G), g ∈ I(F ) ⇒ f g ∈ I(F )”
is obvious. We have now to show I((a)) = a and (I(F )) = F for an ideal a ⊂ O(G) repr. a filter F ⊂ D+(G). The inclusions a ⊂ I((a)) and (I(F ) ⊂ F are obvious. Now, if f ∈ I((a)) \ {0}, then (f ) = (g) with some g ∈ a, hence f = ug with a unit u ∈ O(G) resp. f = ug ∈ a. On the other hand, take a divisor D ∈ F , write D = (f ) with some function f ∈ O(G).
But then f ∈ I(F ) resp. D = (f ) ∈ (I(F )).
4.3 Maximal ideals
Now we are going to describe maximal ideals of the ring O(G). The maximal ideals are also described with help of divisors. We will see that the free maximal ideals have a simpler structure than the fixed ones. Further we consider the factor rings O(G)/m for both free and fixed maximal ideals.
We start to recall some results from algebra.
Let N be an ideal of a ring R. Then the subsets a + N = N + a = {a + n|n ∈ N } of R called the additive cosets of N form a ring R/N with the binary operations defined by (a + N ) + (b + N ) = (a + b) + N and (a + N )(b + N ) = ab + N . This ring is called the factor ring of R modulo N .
The fundamental homomorphism theorem for rings says that if φ: R → R0 is a ring homomorphism with kernel N , then the map µ: R/N → φ(R) given by µ(x + N ) = φ(x) is an isomorphism. If γ : R → R/N is the homomorphism given by γ(x) = x + N , then for each x ∈ R, we have φ(x) = µγ(x). See [1 p.330].
An ideal m of R is maximal if and only if R/m is a field [1 p.336].
We also recall that every integral domain D can be enlarged to a field F such that every element of F can be expressed as a quotient of two elements of D. F is called the field of fractions of D, see [1 section 5.4].
Definition Let F be a field. A F -algebra A is a ring A such that:
1. A is a vector space under addition.
2. k(ab) = (ka)b = a(kb) for all k ∈ F and a, b ∈ A.
A F -algebra A is called a division algebra if it as a ring is a division ring.
Definition Let A and B be F -algebras. A function f : A → B is an F - algebra homomorphism if it is a ring homomorphism and if f (ka) = kf (a) for all a ∈ A and k ∈ F .
From the last definition we see that the elements in C stay fixed under a C-algebra homomorphism.
Theorem 4.13 Let m be an ideal in O(G). Then the following statements are equivalent:
1. m is a closed maximal ideal in O(G).
2. There exists a point c ∈ G such that m = {f ∈ O(G) : f (c) = 0} = O(G)(z − c).
3. There exists a C-algebra homomorphism φ: O(G) → C with kernel m.
Proof 1) ⇒ 2) Since m is closed, m = O(G)f and since m is maximal, m is prime. This implies that f is irreducible, which means that f only has one zero of order 1.
2) ⇒ 3) The evaluation map φc : O(G) → C defined by φc(f ) = f (c) is a homomorphism with kernel m.
3) ⇒ 1) Let φ: O(G) → C be a C-algebra homomorphism, then z ∈ O(G) implies φ(z) = c for some c ∈ C. Therefore φ(z − c) = 0. Suppose c /∈ G, then z−c1 ∈ O(G) and we would have 1 = φ(1) = φ(z−c1 )φ(z − c) = 0.
Hence c has to belong to G. Now let f ∈ O(G) and write f = f (c) − (z − c)f (z)−f (c)
z−c . Since φ keep complex numbers fixed and φ(z − c) = 0, it follows that φ(f ) = f (c). That is, φ = φc where φc is the evaluation map. By the fundamental homomorphism theorem, O(G)/m is isomorphic to C, where m = {f ∈ O(G) : f (c) = 0} is the kernel of the map µ : O(G) → C . Hence m is maximal and clearly closed.
Corollary 4.14 If m is a maximal fixed ideal, then the residue field O(G)/m is isomorphic to C.
Proof Since φc is onto O(G), O(G)/m is isomorphic to C.
The maximal ideals that are not closed, i.e. free, have a more compli- cated structure than the closed and hence fixed ones. We will now charac- terize the maximal ideals, both fixed and free. From proposition 4.12 we see that an ideal m ⊂ O(G) is maximal if and only if M := (m) is a maximal filter in D+(G).
Proposition 4.15 A filter M ⊂ D+(G) is maximal iff min(D0, D) > 0, ∀ D ∈ M =⇒ D0 ∈ M .
Proof ⇒) Suppose M is maximal and the given condition is not valid, then D0 would together with the elements in M generate a filter properly containing M . Hence M would not be maximal.
⇐) Suppose M is not maximal, that is, there exists a filter N ⊃ M . Then for any D0 ∈ N and D ∈ M we have min(D0, D) > 0, hence the given condition implies that D0∈ M , i.e. N ⊂ M and N = M .
To a divisor D ∈ D+(G) we associate its reduced divisor red(D) defined by
red(D)(z) :=
( 1 if D(z) > 0 ; 0 otherwise.
A filter F ⊂ D+(G) is called reduced, if D ∈ F =⇒ red(D) ∈ F ; we define red(F ) ⊃ F to be the unique reduced filter in D+(G) with |red(F )| = |F |.
Proposition 4.16 The filter F ⊂ D+(G) is maximal iff red(F ) = F and A ⊂ G discrete, A ∩ B 6= ∅, ∀B ∈ |F | =⇒ A ∈ |F |.
Proof ⇒) Since red(F ) ⊃ F in any case, the maximality of F implies red(F ) = F . Assume then that A /∈ |F | would be a discrete subset, such that A ∩ B 6= ∅ for all B ∈ |F |. Then ˆF = {D : |D| ⊃ A ∩ B for some B ∈ |F |}
would be a filter containing F as proper subfilter.
⇐) Assume min(D0, D) > 0 for all D ∈ F . Then |D0| ∩ |D| 6= ∅ for all D ∈ F , hence |D0| ∈ |F |, i.e. |D0| = |D00| for some D00 ∈ F . But then D0≥ red(D0) = red(D00) ∈ red(F ) = F , whence D0 ∈ F .
Note that since the support of a divisor is the zero-set of the associated function a maximal ideal m is free if ∩|D| = ∅, D ∈ (m).
Theorem 4.17 If m is a maximal free ideal, then O(G)/m contains the set of all rational functions C(z) as a subfield.
Proof By corollary 4.11, m can not contain a non-zero polynomial. There- fore the composition C[z] ,→ O(G) → O(G)/m is an injective map from C[z] → O(G)/m, that is C[z] is isomorphic to a subring of O(G)/m. Be- cause m is maximal O(G)/m is a field. Then the subfield C[z] can be enlarged to the field of fractions C(z).
Corollary 4.18 The field C is a subfield of O(G)/m. Hence the residue field O(G)/m of a maximal free ideal may be considered as a division algebra containing C as a proper subfield.
In the remaining part of this section we investigate O(G)/m more closely.
The next two lemmas are going to help us. From now on let G = C. First we note that:
1. If (f ) = (g) then f = gh where h is a unit (a nowhere vanishing function).
2. If {zn}∞n=1 is any closed, discrete subset of C and if {wn}∞n=1 is any sequence of complex numbers then there is an f ∈ O(C) such that f (zi) = wi, i = 1, 2, ....
3. A field F is called algebraically closed if every non-constant polynomial in F [T ] has a zero in F .
Let CN be the set of all complex-valued sequences.
Lemma 4.19 The field O(G)/m is algebraically closed.
Proof Let h ∈ m be a function with (infinitely many) simple zeros, say a0, a1, a2, .... Consider the following factorization of the quotient map:
O(G) → O(G)/(h) → K := O(G)/m.
Now the map O(G) → CN, f 7→ (f (an)) has kernel (h) and is, according to 2), onto, hence O(G)/(h) ∼= CN and K ∼= CN/ ˜m with a maximal ideal
˜
m ⊂ CN. Let p ∈ K[T ] be a non-constant monic polynomial and lift it to a monic polynomial q(T ) = (qn(T ))n∈N ∈ (CN)[T ] ⊂ (C[T ])N. In particular, all the polynomials qn(T ) ∈ C[T ] are monic of the same degree > 0, so for any n there is a zero bn ∈ C of the polynomial qn. Then the image in K of the sequence (bn)n∈N is a zero of p(T ). So CN/ ˜m and hence O(G)/m is algebraically closed.
Lemma 4.20 The cardinality of O(C)/m is equal to the cardinality of the continuum.
Proof The dense subset A := Q(i)[z] of O(C) is countable. Consider the map ξ : AN → O(C), where AN is the set of all sequences (fn)n∈N with fn ∈ A and ξ((fn)n∈N) := 0 if fn is not compactly convergent and ξ((fn)n∈N) := limn→∞fn else. Since ξ is surjective and |AN| = |R| = the cardinality of the continuum, we have |O(C)| ≤ |R|. Hence O(C)/m has as most |R| elements. But all complex numbers are incongruent (mod m), so O(C)/m has at least |R| elements. Therefore O(C)/m contains precisely
|R| elements.
From algebra we recall that a field F is an extension field of K if K is a subfield of F . A subset S of F is said to be algebraically independent over K if for all n > 0, f ∈ K[x1, ..., xn] and pairwise distinct s1, ..., sn, f (s1, ..., sn) 6= 0 implies f = 0.
A transcendence base of F over K is a subset S of F which is alge- braically independent over K and is maximal in the set of all algebraically
independent subsets of F . Moreover, every transcendence base of F over K has the same cardinality [8. p.315]. The cardinality |S| of any base S is called the transcendence degree of F over K and is denoted tr.d.F/K.
In [8. p.317] it is proved that if F1 and F2 are algebraically closed field extensions of the field K1resp. K2 with tr.d.F1/K1=tr.d.F2/K2, then every isomorphism of fields K1 ∼= K2 extends to an isomorphism F1∼= F2.
We are now able to state the final theorem of this section.
Theorem 4.21 Let m be a free maximal ideal, then O(G) is isomorphic as a ring to C.
Proof Since O(G)/m contains C it has tr.d over Q at least |R| and by lemma 4.20 it has tr.d at most |R| over Q. Hence O(G)/m has tr.d |R| over Q. The set of complex numbers also have tr.d |R| over Q, and both C and O(G)/m is algebraically closed over Q. Hence by the above O(G)/m ∼= C.
If we replace the integral domain O(G) with any integral domain with greatest common divisor in which Wedderburn’s lemma is true, the principal ideal theorem is valid.
Moreover, for non-compact Riemann surfaces the theorems of Weierstraß and Mittag-Leffler are also valid and hence are Wedderburn’s lemma and the existence of greatest common divisors. Therefore, if we replace regions in C with non-compact Riemann surfaces, all the results in the previous section remain valid.
