In an earlier paper, the author showed that if M is any maximal ideal of R, the residue class field R/M is isomorphic with the complex field K. In this paper, under some restrictions, this theorem is extended to the ring Rλ of all entire functions of order no greater than λ, and hence to R*

Henriksen, Melvin

English

Scholarship@Claremont

Claremont CollegesScholarship @ ClaremontAll HMC Faculty Publications and Research HMC Faculty Scholarship1-1-1953On Rings of Entire Functions of Finite OrderMelvin HenriksenHarvey Mudd CollegeThis Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusionin All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contactscholarship@cuc.claremont.edu.Recommended CitationHenriksen, Melvin. "On rings of entire functions of finite order." Journal of the Indian Mathematical Society 17 (1953): 59-61.58 T. V. LAKSHMlN AHASr:;IHAN ItEFERKNCES 1. n. . .P. lJoAs: FlmdtlUlont,.'ll scts of entire funotions, An1Wls of MatI!. 47 (194G), 21·32. 2. R. O. BUCK: On tho distribution of the zeros of '111 entire function, J01lr. In{U"n Math. Soc. 16 (1952), 147·149. 3. S.)f. SITAR: A note on uniqueness sets for entire functions, ProG. lrulia.n Awd. Sci. 28 (1948), 1·8. 4. S. i\L SHAH: The uHLximllm term of an entire serios (III), Qnartcrly .J01". il[all" .I9 (i9·i8), 220·223. :l!ndras Christian CoIl£'ge Tnmbo.rn::u " i I i I I I I I I I I I t, ON RINGS OF ENTIRE FUNCTIONS OF FINITE ORDER By M {~ININ HENRTICSlm [Received Maroh 26, 1953] In 1940, Helmer showed [I, Theorem 9] that in the ring B of entire functions, every finitely generated ideal is prinoipal. That is, iff. iJ M'e entire functions without zoro" in common, Uwrc exist 8, t in B sueh that sf +tg = 1. (I) He u.skml if this theorem is true in the ring B* of all entire functions of finite ordol'. A nego.tive answer to this question existed >I]wady itl 1930 in a p"per of Whittaker [7, p. 256]. In particular if tho ztl1'OS (a"l, (hal of f, f/ respeotively are not sufficiently sepamted as n -;. 00, (;he equation (I) ca.nnot hold with s, t in B". In 1940, ll1itking usc of results of [7], Mursi showed [6] that if thcre is an 1, > ml1x (ord f, ord f/) such that the eircles S(a •• 'a. i-h) with center "" and rooius I an I-h intersect nonc of the correspondil.g circles 8(b,., I b", !-"), then (1) holds with both ord 8 a.nd ord t no greater than max (ord f, ord g). In an earlier paper [2], the author showed that if M is any maximal ideal of R, the residue class field BIM is isomorphic with the complex field K. In this paper, under some restrictions, this theorem is oxt.endcd to the ring B, of all entire functions of order no greater than '\, and hence to B*' Dm;WlTION. Let i.A (f, n) be. the number of zeros of f contained in Stall' I an 1-'), where a; zero of multiplicity m is counted m times. 'l'rmoRllM. Let -,,,[, be a mtucimal ideal ofR, c(>ntaininll a function f s'Uch that il; (f, n) is bo'Unded for 80me it. 'l'ken BAlM', is isomorphic with 1(. GO ~mI,vm IIENHlKSEN The proof proceeds ItS in [2, Theorem 6). It is eusily seen th,,!, R,!i1:!' has degree oftranscondcncy c (where cis the cardinal number of tho continuum) over the rational field. By (1 well·known theorem of Stcinitz, it is only necessary to show that Ri.lVl' is algebraically closed. Since a maximal ideal of any integral domain is primo, there is a g in if' such that i., (g,n) = 1, for all n. In particular, all the zeros (b,,) of g arc simple. Let (z, X) = X'" + f, (z) xm-l ..L ... + fm(z) be a polynomial with coefficients in R,li1[', of degree m > O. }'or eaoh n, (b", X) is a polynomial with coefficients in K, whioh h"" m complox roots. Choose any snch and call it 0 •. It is well known that ISnl < 1 ..L max (If 1 (b,,) I, ... , 11m (bn},l· Since the ardor of the Ii' and the exponent of convergence of (bn) do not exceed A, it follows from a theorem of Macintyre a.nd Wilson [5, Theorem,,] (also obtained independently by Leont'ev [4,]) that there is a t in R;. suoh that I(bn) = on' So (z, t(z) ) 18 in 111 " whenoe the theorem. REJI'IARK8: L The author does not know if thero is a maximal ideal in R, that fails to satisfy the hypothesis of the theorem. 2. There exist prime idoals of R, and R* that fail to satisfy this hypothesis. :1"01', the set B of elements off of R, (or R*) with i, (f, n) bounded for some h, is closed lmder multiplication. Hence, one can construct, with the aid of Zorn's lemma, primc ideals not intersecting B. Sce also [3]. 3. If, in the theorem R, is replaced by R*, the constant " in the definition of i. (f, n) can be replaced by a positive, increasing function of : z I such that lim sup log h ~I anU is finite. See [5, .... 00 log ,ani Theorem (j J. REFERENCES .. 1. . O. HELMER: DivisibHity properties of integral functions, Dulce J1i at/,. JOUT. 6(1040), 34Jl·356. 2: M. HENIUKSEN: On .the ide",1 structure of· tho ring of entire functions, Pad{1I) JOUT. Matll. 2(1952), 179·184. I , I ON RINGS Ol? ENURE FUNCTIONS OF J?IN1'l'E ORDElt 61 3. M. HE;<f[tI!{SEN: On the prime ideals of tho ring of ~ntire functions, to appear, Pacific Jour. Math. 3(1953). 4. A. F. r."ill:ON'l"II.1V: On interpolation in the class of enUre functions; of finite oeder (n1llmiall), Doklaay AIcad. Nauk. SSSR (N.S.) G1(JIl·18),7%.787. 5. A. J. M"OINTYltE and R. WILSON: On\the order of interpolated intcgrai functions and of IUoromorphio functions with given poles Quart. Jour. Math. 5(193-1), 211·220. . 6. M. MURS!: An itlentity in inkgralfunctiollil, Proo.J1iath. Pllys. Soc. Egypt 1, no. 4(1940), 14·16. 7. J.!VI. WHITTAKER: A theorem on meromorplIie functions, PrO!). London Math. Soc. 40(1936), 255·272. PurdUl) TJrriv(}l'siLy Lt\fayotte", Indiaun, U. S. A.

On Rings of Entire Functions of Finite Order

Keck Graduate Institute

Claremont Colleges
Scholarship @ Claremont
All HMC Faculty Publications and Research HMC Faculty Scholarship
1-1-1953
On Rings of Entire Functions of Finite Order
Melvin Henriksen
Harvey Mudd College
This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion
in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact
scholarship@cuc.claremont.edu.
Recommended Citation
Henriksen, Melvin. "On rings of entire functions of finite order." Journal of the Indian Mathematical Society 17 (1953): 59-61.
58 T. V. LAKSHMlN AHASr:;IHAN 
ItEFERKNCES 
1. n. . .P. lJoAs: FlmdtlUlont,.'ll scts of entire funotions, An1Wls 
of MatI!. 47 (194G), 21·32. 
2. R. O. BUCK: On tho distribution of the zeros of '111 entire 
function, J01lr. In{U"n Math. Soc. 16 (1952), 147·149. 
3. S.)f. SITAR: A note on uniqueness sets for entire functions, 
ProG. lrulia.n Awd. Sci. 28 (1948), 1·8. 
4. S. i\L SHAH: The uHLximllm term of an entire serios (III), 
Qnartcrly .J01". il[all" .I9 (i9·i8), 220·223. 
:l!ndras Christian CoIl£'ge 
Tnmbo.rn::u 
" i 
I 
i 
I 
I 
I 
I 
I 
I 
I 
I 
I 
t, 
ON RINGS OF ENTIRE FUNCTIONS OF 
FINITE ORDER 
By M {~ININ HENRTICSlm 
[Received Maroh 26, 1953] 
In 1940, Helmer showed [I, Theorem 9] that in the ring B of 
entire functions, every finitely generated ideal is prinoipal. That is, 
iff. iJ M'e entire functions without zoro" in common, Uwrc exist 8, t 
in B sueh that 
sf +tg = 1. (I) 
He u.skml if this theorem is true in the ring B* of all entire 
functions of finite ordol'. A nego.tive answer to this question existed 
>I]wady itl 1930 in a p"per of Whittaker [7, p. 256]. In particular 
if tho ztl1'OS (a"l, (hal of f, f/ respeotively are not sufficiently sepamted 
as n -;. 00, (;he equation (I) ca.nnot hold with s, t in B". In 1940, 
ll1itking usc of results of [7], Mursi showed [6] that if thcre is an 
1, > ml1x (ord f, ord f/) such that the eircles S(a •• 'a. i-h) with center 
"" and rooius I an I-h intersect nonc of the correspondil.g circles 
8(b,., I b", !-"), then (1) holds with both ord 8 a.nd ord t no greater 
than max (ord f, ord g). 
In an earlier paper [2], the author showed that if M is any 
maximal ideal of R, the residue class field BIM is isomorphic with 
the complex field K. In this paper, under some restrictions, this 
theorem is oxt.endcd to the ring B, of all entire functions of order 
no greater than '\, and hence to B*' 
Dm;WlTION. Let i.A (f, n) be. the number of zeros of f contained 
in Stall' I an 1-'), where a; zero of multiplicity m is counted m times. 
'l'rmoRllM. Let -,,,[, be a mtucimal ideal ofR, c(>ntaininll a function 
f s'Uch that il; (f, n) is bo'Unded for 80me it. 'l'ken BAlM', is isomorphic 
with 1(. 
GO ~mI,vm IIENHlKSEN 
The proof proceeds ItS in [2, Theorem 6). It is eusily seen th,,!, 
R,!i1:!' has degree oftranscondcncy c (where cis the cardinal number 
of tho continuum) over the rational field. By (1 well·known theorem 
of Stcinitz, it is only necessary to show that Ri.lVl' is algebraically 
closed. 
Since a maximal ideal of any integral domain is primo, there 
is a g in if' such that i., (g,n) = 1, for all n. In particular, all the 
zeros (b,,) of g arc simple. Let (z, X) = X'" + f, (z) xm-l ..L ... 
+ fm(z) be a polynomial with coefficients in R,li1[', of degree m > O. 
}'or eaoh n, (b", X) is a polynomial with coefficients in K, whioh 
h"" m complox roots. Choose any snch and call it 0 •. It is well 
known that ISnl < 1 ..L max (If 1 (b,,) I, ... , 11m (bn},l· Since the ardor 
of the Ii' and the exponent of convergence of (bn) do not exceed 
A, it follows from a theorem of Macintyre a.nd Wilson [5, Theorem,,] 
(also obtained independently by Leont'ev [4,]) that there is a t 
in R;. suoh that I(bn) = on' So (z, t(z) ) 18 in 111 " whenoe the theorem. 
REJI'IARK8: L The author does not know if thero is a maximal 
ideal in R, that fails to satisfy the hypothesis of the theorem. 
2. There exist prime idoals of R, and R* that fail to satisfy this 
hypothesis. :1"01', the set B of elements off of R, (or R*) with i, (f, n) 
bounded for some h, is closed lmder multiplication. Hence, one 
can construct, with the aid of Zorn's lemma, primc ideals not 
intersecting B. Sce also [3]. 
3. If, in the theorem R, is replaced by R*, the constant " 
in the definition of i. (f, n) can be replaced by a positive, increasing 
function of : z I such that lim sup log h ~I anU is finite. See [5, 
.... 00 log ,ani 
Theorem (j J. 
REFERENCES 
.. 
1. . O. HELMER: DivisibHity properties of integral functions, Dulce 
J1i at/,. JOUT. 6(1040), 34Jl·356. 
2: M. HENIUKSEN: On .the ide",1 structure of· tho ring of entire 
functions, Pad{1I) JOUT. Matll. 2(1952), 179·184. I , 
I 
ON RINGS Ol? ENURE FUNCTIONS OF J?IN1'l'E ORDElt 61 
3. M. HE;<f[tI!{SEN: On the prime ideals of tho ring of ~ntire functions, 
to appear, Pacific Jour. Math. 3(1953). 
4. A. F. r."ill:ON'l"II.1V: On interpolation in the class of enUre functions; 
of finite oeder (n1llmiall), Doklaay AIcad. Nauk. SSSR (N.S.) 
G1(JIl·18),7%.787. 
5. A. J. M"OINTYltE and R. WILSON: On\the order of interpolated 
intcgrai functions and of IUoromorphio functions with given poles 
Quart. Jour. Math. 5(193-1), 211·220. . 
6. M. MURS!: An itlentity in inkgralfunctiollil, Proo.J1iath. Pllys. Soc. 
Egypt 1, no. 4(1940), 14·16. 
7. J.!VI. WHITTAKER: A theorem on meromorplIie functions, PrO!). 
London Math. Soc. 40(1936), 255·272. 
PurdUl) TJrriv(}l'siLy 
Lt\fayotte", Indiaun, U. S. A.

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On Rings of Entire Functions of Finite Order

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