A STR.CT1RE THEORY 0F MODULES
WITH A SPECIAL REFERENCE TO THOSE
DEFINED BY THE CMtHY CONVERGENCE CRITERION
by
James Russell Brown
A THESIS
submitted to
OREGON STATE CCLLEGE
in partial fu1fillrnnt of
the recwireitents for the
degree of
MASTER OF AHTS
June
l98
APPROVED
Associate Professor of Mathematics
in Charge of Major
Chairman of Department of Mathematics
Chairman of School of Science Graduate Committee
Dean of Graduate School
Date thesis is presented
Typed by
:--(
ever1y Saralyn Brown
IC
I95-
TABLE OF
CONTTS
Pa go
INThODtXT ION
SECTION
DefinitIons
1.
arid
Preliminery Results
3
SECTION 2.
Complete Normed Modules
lo
SECTION
Rings and Ideals of Bounded Secuences
25
Other Ringe Related +n the Real Number
Field
29
3.
SECTION 4.
BIBLIOGRAPHY
34
A
STRtLTURE THEORY OF MODULES
WITH A SPECIAL REFERENCE TO THOSE
DEFINED BY THE CAIXHY CONVERGENCE CRITERION
INTROD(XT ION
familiar construction of the re?1 numbers as
The
ecuiva1ence classes of Cauchy sec'uences
assumes
particularly elegant
a
of
rational
numbers
form when expressed In
alge-
A development of this type is given, for
braic terms.
instance, by Van der
be the rational
v:aerden [5, p. 211-2171.
field,
number
let
R
the set of Cauchy se-
the set of null secuences
define addition and multiplication of
ouences of elements of R, and
of elements of R.
R1
Thus
We
N
elements in R1 by
[a}
+
b}
[a
4
=
Then R1 is
commutative ring with identity and
a
1mal ideal in R1.
field, called the
real
number field.
numbers, an
into R1/1.
The principal results
(i) R1/N
a
o
field;
subfield R' of
is a max-
Hence the residue class ring R1/N is
sec'uences of rational
is
N
(2) R is
Rj/;
(3) R'
By defining "positive"
ordering is
concerning
ring-
a
introduced
R1,44
are:
and order-isomorphic
is dense in R14';
complete, that is every Cauchy secuence
of
(4) R114
to
is
elements of R1,4
converges; and (5) every bounded set of elements of Rj/
has
a
least upper bound.
These results may be extended without any groat diffiCulty to the case wher
(Archjmedean or not)
sent
R is an arbitrary ordered field
[5,
211-2171.
p.
investigation is two-fold:
The aim of the pre-
first, to vary the above
construction by considering rings of seruences of rational
numbers other than R3 and kernels other than N and
to in-
vestigate the algebraic properties of the ouotient rings so
obtained; and, secondly, to generalize the construction and
the above-mentioned results to other metric-algebraic varieties, in such a way as to include some results which are
known, as well, possibly, as some which are not known.
In
results
section
1
we state and prove some preliminary
later sections. Section 2 Is inpartial solution of the second problem
which are used in
tended as at least
a
posed above, while sections
and
intent, treat
some
3 and 4, algebraic in nature
very special cases of the
first
problem.
Throughout
this paper free use is
made of such
well-
known results as the "fundamental
theorem of homomorphism
for groups with operators"
133
tions to rings
[2,
and to modules.
p.
and its specialize-
In addition are used the
following set-theoretic and algebraic symbols, with the
usual meaning:
Finally,
X,
c,
,
(L
,
we adopt the abbreviation
jx,)
(denoting
"1ff" for "if
a
ard
se6uence.
on' if."
ci
SECTIJ 1.
DEFINITIC*S
AND PRELIMINARY RESULTS
this section with some fundamental theorems
from the thory of rings and ideals.
be arbitrary rings such that
Theorem 1. Let P and
P
and let N be an ideal in Q. Then the intersection
N(\P is an ideal in P.
Proof. NP is not empty, for C NfP. If N/P
If x,y c N('tP and p e P,
(o), then NIP is an ideal in P.
then x - y P and x - y t N, so that x - y t Nfl?. Also
We
begin
(
C
X
P,
plies
implies xp P, px
e O, which implies xp c
N(tP.
p
and px
N, p
P; and x
p L P
N, px
Thus xp
N.
im-
P
NCIP
Corollary 1.1. If N, P, Q are rings such that N P
Q and N is an ideal in Q, then N is an ideal in P.
are rings such that P Q and if
Theorem 2. If P and
N is a prime ideal in
then N(\P is prime in P.
Proof. Suppose x,y t P and xy e Nt'P. Then x,y L Q
and xy e N. Since N is prime in O, either x e N or y e N.
C?
Hence
either
If
A
and
x e
B
N/P or
y e
N,IP, and NR? is prime in P.
are subsets of
a
group R,
we
denote by
A + B
the set of all elements a + b of R, where a e A and b e B.
In general, A + B will not be a ring whenever A and B are.
However, we have the following theorem.
Iheor 3. If P and Q are rings such that P C and if
Moreover,
N is an ideal in Q, then P + N is a subring of C.
it is the smallest subring of O containing P and N.
4
Proof. P + N is certainty not empty. Supr.ose
n2 are arbitrary elements of P + N with P1,P2
+
P2
P
+
n1,
and
Then
N.
(Pi
p3.
1)
=
(p2
-
(p -
p2)
+
(n1
fl2)
E
+ N
P
and
(pj nj)(p
Since N is an idoal
+ fltP2
+ fljfl2
that
+ N
Ing both
p
+
P
n c R
is
Q,
a
P
ring.
N. If
-f
N
p
L
C R.
fliP2
L
N, so
Since
pjp2
L
P,
let
Now
P
that P1fl2
this proves
and
Pifl2
belongs to N.
end
and
= (p1P2) + (pin2 +fljp2 tfljfl2).
R
be
a
subring of
and n s N, then p,n
contain-
Q
E
Hence
R.
This corrpletes the proof.
If N is an ideal in R, we write R/N for the residueclass (ouotient) ring of R modulo N. Now In the above
theorem we have, by corollary 1.1, that N is an ideal in
p +
N
.
Moreover,
if
o s P +
N
and we denote by
and
,
respectIvely, the residue classes in Q and in P + N which
However, since any
.
contain o, then ouite obviously
n where n s N,
may be written in the form
element in
. Thus ve arrive at the following inportant
we have
4-
isomorphism theorem.
heorern 4.
If
P
and
O
are rings such that
P
Q
and
N
(P +N),k
Q/.
ideal in C, then P,'(PiN)
s QA
be the residue
let
X
Proof. For ec
class containing x and consider the mapning x -e- of P into
(P+N),ì. Clearly this mapping is a homomorphism. Moreover, it is onto, for any element of P + N may be titten
is
an
'
5
in the form p + n with
x
P
e
and
then
,
p e
x e
and n
P
N
N,
and hence
kernel of the mapping is PCN and
by
and p + n =
P\N.
If
.
the
the fundaienta1 theox e
Thus
rem of homomorphism
P/(PrN)
have already seen
We
that
(P +N)A
If
and
the theo-
Q/N, and so
is proved.
rem
4.1.
QLol1ary
in
such
,
that
of
subring
is
extension of
O
Q
are rings and
ideal in the ring
such th3t P/IN
if
Q/
is
N
ideal
an
Q/.
then P,4
If N is an
P
.
P
Q,
P
N
Corollary 4.2.
P
(p +N),4.
(0), then
and only
if
P
O4
,
and if
is an
intersects every
residue class of modulo N.
Next we introduce the concept of a module and give a
brief re'sum of the elementary theory.
Definition 1. Let A be a ring. A set
is an A-mod.LkQ 1ff R is a commutative group (operation +) and there
C)
exists
a
on A X R to R such that for all a,b
function
and all x,y
1)
ii)
e
R,
a(x
(a
+y) = a.x
+
a'y.
ax
f
bx,
+b).x
=
iii) (ab)x a'(b'x).
the "productt' ax we shall write briefly
For
Definition 2. Let
is
a
submodule of
i)
A
L
R
R
be an A-module.
iff for all x,y
e
N
A
ax.
subset
and all.
N
a e A,
of
R
ifl axN.
Nw
let
x
y
be an A-module and
R
iff
partitions
x
R
y,
y and u
submodule of R.
e
define
is an ecuivalence relation and
into eruivalence classes . Furthermore, if
- y
x
a
N
Then '
e N.
have
we
x + u
y
+
y and ax
ay
for all
are thus led to make the fo1lowin definition.
DefInition 3. Let R he an A-module and N a submodule
of R. The A-module consisting of the set of ecmivalence
classes defined by N and the compositions given by
a
A.
We
i)+x+
forallx,yER,
for all x e R, a e i, where
;
the eriulvalence class containing x, is called the
Qtient moduLe of R modulo N, and is denoted by R/.
That R/ is indeed an A-module is seen by the following eouations:
a(x+y)
a( f) =
ii)
ax
=
3(T)
(a+b)
(ab)
y
x
of
P
Xj
ii)
atbx)
c;i;T; =
Definition 4.
mapping
axbx
(TTT
x -
a(b).
single-valued
homomorphism 1ff
Let i-,C be A-modules.
onto
'j,
y
a()
X2
C
is called
Y2
implies ax
a
implies
-+
ay for
Xj
A
+ X2
all
+
a e A.
is called the homomorphic image of P. The kernel of the
homomorphism is the set of all x e P such that x-0. If
the mapping is l-1, lt is called an isomprphi and we
C
write P
Q.
7
odu1e
A
concept of
is only
sreciei cese of the
a
more Qeneral
group with operators (seo Jacobson [2, p. 128-
a
186]). Thus the entire theory for the latter carries over
to the former. In particular, we hve the
fjindarnenta1
If
Homomorphism.
florern
P
is
an
I.-
is a submodu1e then is homomorphic to P/a;
and, conversely, if F is homomorphic to R and N is the kernel of the homornor}hisn, then Is a submodule of P nd
module and
G
I
P4\i.
R
L2, p.
(uite obviously if
is
submodule of R, then
state the analogue of
Theore
suhrnodules of
Ci.
a
¿are A-modules and P
submodule of L.
We
is
a
may now
theorem 4 for modules.
Let Ç
5.
P c Q c R
Thon
let
be an A-nodil.e and
P(N and
P +
N
I
and
N
be
are submodules of
L
and
P/(PiIN)
Proof.
P(\N.
ax
e
Also
Pt\N.
Supìose x,y
x
x e
a
submodule.
are arbitrary members of
+ Y2
Y1.Y2 e
arid
N
let
(xi +yi) -
a
A.
Then
(x2
y2)
(xj
+ N
-X2)
Then x - y
a e A.
N implies ax e P,
PIN is
Thus
e
(p+i)/!
PN and
N
ax
Suppose
with
+ (yj
+
x,x2
e
and
a(xj +yl)
Thus
P
-
N
is
a
=
submodule.
ax1
ay1 e P +
and hence
Xj
y2)
N.
e
y
e P + N
and
arid
LJ
Again let
be the e.uiva1enc. class of x
and consider the mapping x
ping is
a
£
A.
a
-
e
of P onto P + N.
module homomorphism since ax
=
-+
Q modulo N
This mapfor all
a
The proof follows then exactly as for theorem 4.
Corollary 5.1.
N Ç P Ç p, then
If N,?
Ç
P4':
Corollary 5.2.
Q/1'.
If Q is an A-module and if N and P are
submodules of C such that
sion of P.
Ç4
P
Q are A-modules such that
arid
(o),
NiP
then ÇA'
is an exten-
iff P intersects every eruivalence
class of C modulo N.
Theorems 4 and 5 may both be deduced from the "second
isomorphism theorem" for groups with operators as stated
and proved in Jacobson L2, p. 136].
following theorem is
phism theorem" [2,
Theorem 6.
if M
Ç
135].
p.
If i! and N are ideals in the ring P, and
For each x
e
P,
and
P,41
p/.
(P4)/(N4)
Proof.
the
special case of the "first isomor-
a
then N/M is an ideal in
N,
In a similar way,
let
L
P4QI
and
L
P/
be the
ecuivalence classes modulo M and N respectively which con-
Consider the ma.ing
tain x.
maping
is
hence
-
x
X
x
e
N.
Thus
;
= X
x.y
so that the map::ing
is
a
Moreover,
.
'
P4 onto
Ç, then
single-valued, for if
y
of
+y
xy
-+
+,
-xyxy,
homomorphism.
Now
x
P/Ì'J.
-
y
The
arid
1ff zeN 1ff
so that N,' is the kernel of the homomorphism. By the
fundamental theorem of homomorphism then, N4 is en ideal
P/N.
and (P/M)/(NA1)
if rings
This theorem obviously remains velid
and
ideals are replaced throughout by modules end sub-modules.
However, we shall riot make use of this fact in the secuel.
Theorem 7. Let Q be a commutative ring with identity,
r
and let MO be the set of all elements of the form E m.o.
i=1 '
O, nj e O (11,2,...,r). Then MO is the
where m1 M
smallest ideal in Q which contains M.
s
r
If
Proof.
Z
m11
e
MC
and
E mId
e MC,
and 1f
1=1
1=1
t h en
r
r+s
s
in
o.
E
1=1
m!c
1
rn"o'.'
1
*
1=1
and
r
r
E
i
L
i=l
is an ideal In O, Since
r
o E
Thus
MO
for in e M Implies me
ideal in C and M P, then
MO,
in
Mfl
E
1=1
Q
m.o
1 L
identity e, M
Finally, if P is an
has an
e MC.
PC =
e MO.
P.
o
2.
SkCTION
As
before
hi
w'
COMPLETE NORMED
by a
me&r
ODTJLES
complete system or space
one in which every Cuchy secuence converges.
shown
Metric
that
It my
be
any metric spece can be imbedded In a complete
SpaCe,
re eouiv1ence classes of
whose eiements
seuences [4, p. 84-881. The space which we shBll
Consider is fl sorne ways 1es genri, but it includes as
speciel czses two im:ortant applicatlons of this construction integral domains with valuation and norrned linear
spaces. toreover, the present theory has the adventage of
preserving the algebraic structure of these special cases.
Thus an integral domain will be imbedded in an integr'l domain and a linear space in a linear space. We begin this
section wits some preliminary definitions.
Definition 5. A valuation for an integral domain A is
A,
a function ç on A such that for all a,h
i) p(a) P, a completo ordered field,
ii) ç(a) > o for a O; (o) = o,
Cauchy
:
iii)
iv)
p(ab)
c(a +b)
<
Since any ordered field
tp(a)
+
may be
com1.eted in the manner indi-
cated in the introduction, it is no restriction to suppose
that P is complete. We note that, if A has an identity e,
p(-e) = 1. In any case
then by li) and iii) cp(e)
=
11
Recalling the definition of an A-module from the previous section we note that by i) and ii) of definition 1,
1f o E A and O C R are the respective "zero" elements, then
and all
ox = aO = O and (-a)x
a(-x) -ax for all a
R. In case A has an identity e, we shall for the sake
x
of simplicity insist on the additional restriction
iv) ex = x for ali x c R.
We are now justified in making the folloing definition.
Definition 6. Let A be an integral domain with valuation p to P an ordered field. Let R be an A-module satisfying the condition
O
o or x
o) ax = O implies a
for R is a function i on R
for all a E A, x R. A
IA
E
such
that for all x,y
i)
(x)
ii)
i(x)
iii)
=
i(x)
and
ìl
A,
a
P,
> O
O:
for x
O,
ii(o)
.t(ax)
(x+y)
iv)
Since
R
t(3x) =
<
i.t(x)
q-a)p(-x)
by condition iii).
ciled briefly
+
cp(a)i(-x),
A module with
a
we hcve
(-x)
norm defined is
ring A itself is a
normed module if R is taken to be the additive group of A,
multiplication !s ring multIplication and .i = p. If A is
the real (complex) number field and (a)
a1, then R is a
real (complex) normed linear space. In any case, since
a normed
module.
The
12
t(x), R is
*
metric space, with the metric
a
(x,y;
y).
We are now ready to define what is meant by
a
Cauchy
se'uertca in a norrned modulo and thus to introduce the nues-
tion of completeness.
flnitiçm
values in
[x}
and let
Then
of R,
Let R he an
7.
i)
having
be an infinite senuence of elements
is
convergent 1ff there exists an
each positive ¶
n > N
ii)
Amodule with norm
implies
P an
E
x
R and for
integer N such that
<
.L(x -
Cauchy 1ff for each positive
P
'r
there exists
an integer N such that
n.m > N
iii)
irplios
(xx)
null 1ff for each positive
an integer
n > N
In case i) we call
x
1.1
'r
<
'r;
e
P
and
there exists
such that
implies
(x)
the Limtt of {x}
<
'g.
tt. x
11*
(or 11m xe).
n
Let R be an arbitrary norred A-module (norm in ?);
N be the set of null senuence
of elements of R; R0 the set
of convergent senuences of elements of R; and R
Cauchy se'uences of elements of R.
R1.
the set of
If R is complete, R
Otherwise, we shall see that R may be imbedded in
complete A-module,
let
Let us defino addition in R
and
a
13
multiplication of eiecrents of R1 by elements of A by
+
[yI
=
{x
a[x)
=
Íax).
We shalt show that N, R0 and R1 are Amoduies and that N
R0
R1.
The associative and commutative laws of addition follow immediately from the corresponding laws in R.
[x}
[X1
+
+
oreover,
[xh
[01
[-X
fol,
where [01 is the secuence in which each element is 0.
addition we have
(a+b)(x)
'4ax3
aUx}
+
Eye))
=
atx1
a(x+y}
=
[ax+ay)
identity,
Now if (x),[y)
-
a[x)
=
a[bx
t
[i.X) m
Rl and
- (xm
c
a
A,
then since
A(x
- Xm)
y
y1)
+
(y
we have
L(
x
+
then
l'[xfl}
Ux
+ (aya)
f(ab)x} r [a(bx)}
(ab)h)
+
+ (t,xI
z[ax)
If A has
[ax+bx}
[(a+b)x)
- xm)
< 'r/2,
(
< i/2
-
in
14
inì pl le s
((x-y
n
)(x m
ymfl<;
and
(xnxm)
<
Implies
p(a)(x_x)
(axnaxm)
provided
a[x}
R1.
-
This shows that
O.
a
Thus R1 is an A-module.
<
't:,
R1 and
[y}
Similarly, R0 and N
are A-modules,
Suppose [x}
each positive
R0.
E
P a
-r
n > N
Then there exists
R and for
x E
positive Integer N such that
i(x-x)
implies
< -/2.
Thus
m,n
and
> N
[X)
{ x}
t
then 11m
8
Proof.
Let
fines
a
R
R oA'
4x -
xm)
submodule of R3.
a
t
R0.
Thus N is
1
that R014
<
'r
If
a
sub-
and R1/14 are also
a/N.
be the A-module of secuences
= X for all n,
Then R
l-1 mapping of R onto
R and all a
+
They are related in the fo11o;;ing manner.
ThtcL
x
-
R,,
We have seen in section
which
(x
= O and Ext)
module of R0 and hence of
A-modules.
<
xm)
Therefore, R0 is
R1.
N,
i(x -
implies
e
A,
?
Ro, for
[xfl}
x -
[x
for
de-
under which, for all x,y
t
15
+y
ax
Since [x} converges to
Now(ÌN
X
for each
[o), and
Thus R
L
Proof.
1,
If
N.
By
By corollary 5.1,
R/
If R Is complete,
e
then R14
'
R.
is complete.
R1, then [R(x)) converges.
e
{xr,)
[x}
Ç
Next we show that R114
Lemma
R0 there exists
that Is (xe)
= x,
R0/t4
Qr.pi1ary 8.1.
[y),
R0.
corollary 5.2, we have
R1,43.
+
[ax)
-
x,
R such that 11m x
E
[x}
y)
[x
From the properties of
<
i(x
xm) +
.t(X)
<
a
norm,
i(x..x).
Also
t(xm)
L(X-X)
<
-
(x
Xm)
Thus
-
Therefore,
[.(x))
is
a
(x)I
A(xn_Xm)i
Cauchy secuence of elements in
Since P is assumed to be complete, the lemma follows.
Let
*([))
= 11m
where [;;) is the ecuivalence class modulo N which contains [x}.
P.
16
Lemma 2.
is a norm for
R/
and agrees with
ii
on
the image R0,M of R.
Proof,
First we show that R1/N satisfies condition o)
of definition 6.
either a
O or
a[)
Thus if
{x)
e
N.
then (ax,)
O,
Therefore, either
a
s
N and
O or
= O.
By definition and lemma
a N,
t*({1)
1,
s P.
If (xe)
then
um M(x-y)
O.
But
for all n, so that
lia (L(x)
lia it(x)
and
1.L*
is
lia
singlevalued,
((r})
a O
*((1)
O
ii(y))
(y)
o,
a
Moreover,
iff
[c)
since
s N
iff
(x)
O
end
j*(ax
lia
a
a
J)
a
L(ax)
(a)lim L(x)
t;;i
o,
for ill n;
17
Finally
y)
t(x
i(x)
<
(y).
+
(n
1,2,...),
{x)
-
so that
*([x
+y})
+
t;})
This proves that
Now if
*([})
+
L*([)
+
;
is a norm,
(1
is the image of x t R,
then
e
N
and as above
litri
um
(i(x)
-
(x))
O,
*U'})
(x)
This completes the proof of lemma 2.
Lemma 3.
is dense in
Ro,4
R4' with respect
topology imposed by the metric T«x,y)
Proof,
[}
Let
y).
be any element of R1,4
Then there exists
tive element of P,
to the
a
and
't
positive integer
such that
> no
n
implies
i(x
n
-x
)
<
r/2.
o
Thus 1f [yl is the constant seruence defined by y
for all n, then
ri
or,
>
o
(x-y)
implies
<
passing to the limit
*((x_;))
*([}
..
ty})
¶72 <
<
'V,
any posi-
1,
c/2,
o
p'.]
-[})
;i
But
:
h}
R044, since
r.
converges to
x0.
Thus every
[}
'p-neighborhood of
contains an element of R0ft.
eierrent of R14< is the limit of
Every
Lemma 4.
<
a
con-
vergent seouence of elements of R0/N.
Proof. Let h;} be a monotone, null seouence of positive elements of P. By lemma 3, there exists for each a t
R1,4,
<
a
¶
with
seouence
Thus, for each '
integer
rio
such
n >
> O
that P*(a _ r)
there exists a positive
RoAI such
in P,
that
implies
no
.L*(aj3)
<
<
¶,
and hence
u = 11m
Theorem 9.
is complote with respect to the
a114i
norm
null se-uence of elements of P.
( In the Archimedean case we may take, for instance, '
be a Cauchy secuence of elements of R1/N
1/n.) Let
Proof.
Let
be
secuence of elements of Ro/N such that
j*(
<
and
let
Now
for any positive
and
fl2
be
such
a
¶
t
P
there exist positive integers
that
n,rn
and
a
>
t
implies
i*(a_a)
<
/3
n1
19
T
Then for n,in > n0
.*(p
n
.
m)
max(nj, na), we have
*(_u)
<
+
+
+
+
+ ¶13 + ¶/3
and hence
Let
R.
{}
is
be
By lemma
under this isomorph.sr.
eouonce.
[xL
Then
By theorem 8, R0A1
e1ments
sectaence of
a
T
Cauchy soouence.
a
1L*(am_3m)
'
of R such that
2,
xJ
is a
Cauchy
R1,4 be the eouivalence class containing
Let
for each n
ji*(a,)
L*(_a)
1(Xk -x) +
p.*(c_,) +
hin
k
Now there exists an integer fo such that
irpies
r,k > r.
(xk
-x)
<
i/2
and
fl
>
no
implies
T1.
< ¶/2.
Thus
implies
n > no
11m
L(Xk
-xe)
k
implies
*(a_a) <T.
Thus i:al converges.
We
have shown
that any normed module can be imbedded
in a complete normed module.
pi.x)
In particular, any real (corn-
normed linear space can be imbedded in a
real
20
this section will
devoted to applying this result to integr1 domains.
Theore 10. Any integral domain with valuation can he
(complex) Banach space.
be
imbedded in a complete
Proof.
Let
R
be
having values in P,
The remainder of
integral domain with valuation.
an integral domain with valuation
Then we have seen that
R is an
R-module
the above construction we obtain the cornplete normed R-module R1A. Now since R R04, we have,
in an obvious way, that R1/ is an RoA-moduie, in the sense
* is a valuation for R0,41
of isomorphism. Moreover, p*
and p* j$ a norm for RjA'4 with respect to this valuation.
Let OE,B he arbitrary elements of R1/. By lemma 4, there
exists a seruence
of elements of R0/N such that
with norm
p.
By
11m a
The secuence
ta}
given any positivo
is
r
a
L
for, if
there exists an Integer
Cauchy senuence,
P,
f
0, then
such
tha t
rn,n
>
o
implies
implies
If
ß
= O,
then
_arn)
<
ç*(an_aîn1)
[01
is
a
nm)P*3)
<
Cauchy se-uence.
define two compositions + and in the set R1AT in
the following way. + is the group addition in the module
R1,4. For any a, e R1,4\T
We
a2
wh er e
= 11m
21
1ima. aRo/.
[Note
that if
a
t R0/4, then a!3
11m a$
n
This composition Is single-valued, for
ges to a, then (a
null senuence.
-a}
if [a} also
conse'uently [a
and
shall show that
gral domain.
is
-
a
11m
a3.
with these compositions is an into-
R1/W
We
theory of double
numbers
seruences of real
247-292] is easily carried over to R1/N.
In
p.
E3,
particular, if
are convergent seruonces, then
1%) end
+)
11m
um a nfl
where a
conver-
Thus
um aB
The
afl.]
11m
OEA.
B
= 11m a
11m
n
a n 11m
k
The
= 11m
+ 11m
first
Bk
of these is
a
con-
seruonce of
To
prove the other
note that for
wo
Q*(a)
some
M
C
< M.
Thus
,*(a),*(.)
=
so that (u
-aB} is
a
11m
nul]. seouence.
(a3_aP)
= O
< M
Thus
P
and
all
n
22
Now let
11m
um
1a, {}
be Cauchy secuences of elements of
= 11m a0
R04 converging respectively
of
R1/,
11m
to the arbitrary elements
and let y he any other element of R111.
x
and
Then
= 11m a0(iim
= 11m 11m
n
k
= 11m 11m
n
=
flkh'
k
),Im
-(u')'y.
Also
+ y)
=
lin!
= 11m
cx(
+ y)
(cL
+ay)
+
11m
=
'3
11m
+
and
= 11m
(-e)y
= 11m
(!1y+By)
11m
y +
um
= a.y + B'y.
Fina11y
suppose a'B
and for each positive
'r
O and
P
O.
Then 11m iÇ3
there exists an integer
fl
O
such
23
tha t
> no
n
*()
Since
O,
*()
implies
<
'r
this implies
=
arid hence
um
a
= O.
is automatically a commutative group with respect to addition by the definition of +. Thus R1/Ï is an
integral domain and theorem 10 is proved.
If R has an identity e, then RoA has an identity and
Now R14
=
=
jim
so
is
that
an
ae
=
a,
um
identity for R1/I.
If
R
is commutative,
t h en
=
a3.
um
um
is also commutative.
Finally, if R is a division ring, then
Thus R1/
(11m
80
¿1),a
= 11m
Çc
um
a1k
um tÇ1a
that
a_l
and
11m
w
=
um
have proved the foitowing
field with valuation can be imcomplete field with valuation; any division ring
Cpro1lar
bedded in
a
corollary.
10.1.
Any
24
with valuation can be imbedded in
with valuation.
a
complete division ring
25
3,
SECTION
RINGS AND IDEALS OF BOUNDED SECUENCES
to the investigation of sequences of rational numbers. Let R be the rational number field, let N,
R0, R1 have the same meanings as in the revious section,
and let R2 be the set of all bounded seouences of rational
numbers. Thus N CR0 CR1 CR2. We define addition and
multiplication in R2 just as we did for R1 in the introducW
return
now
tion; thus,
[a}
+
[b}
[a)'[b}
{a
+
b}
[ab}.
is a commutative ring with identity.
Proof. The associative, commutative and distributive
laws follow directly from the corresponding laws for the rational number field. Let [a} be any element of R2. Then
{a} + [o)
Theorem 11.
R2
[aa)
+ [.m.a}
= [O),
[a)'[i}
is the additive identity, [-a} is the additive inverse or "negative" of [a}. and [i) is the ring identity.
Suppose now that [ar), [b} e R2. Then there exist rational
numbers A and B such that
Thus
(o)
IaI
Then
< A,
fbJ
< B
for all n.
26
Ia
+bI
<
[a}
Thus
JbI
<
A +
L3
IaItbI <AB
IabI
for all n,
+
Ia1.I
a}(b}
[b}.
+
R2.
This completes
the proof.
N is an ideal in R2.
Theo;em 12.
Suppose
Proof,
exists
a
By the definition of
> O a
t
{b}, [c}
R2;
N.
Then there
rational number A such that
Ial
'r
c
[ar'
< A
null secuence, there exists for each
a
such that
positive integer
n
> no
Likewise, there exists
n > n1
Theorem 13.
Proof.
implies
lbI
implies
IabI
a
< 'v/A
'r.
<
positive integer n1 such that
implies
IbI
implies
Ib-cI
Ib
t
is an
fb}
Thus 1a}'[bn'
for all n.
{c1}
t/2,
<
N,
nd N
¡cRI
n
I
< 'r/2
+ Ic n
I
<
ideal in R9.
N is not prime in R9.
Consider the secuences
a
(o,i,o,i,...}
b
Now a,b
R2
-
N and ab
[o)
N,
integral domain.
Corollary 13.1.
R9/
Ço1lar.y 13.2.
N is not maximal in R9,
is not an
'r.
27
conclusion that
We are thus led to the
there exists at
least one ring P such that
NCPCR
i)
P is an ideal in R2.
ii)
Let
be the class of all rings P satisfying i) and ii).
JI
By theorem
p
PuR1
is an ideal
PuR1, and since
over, N
Pr\R1
1,
{i}
in R1
£ R1
for all P
and [11
E
' P,
II.
More-
we have
Since N is maximal in R1, this proves the fol-
R1.
lowing theorem.
Theorem 14.
PIR1
Theorem 15.
R24'
N for all P
is
a
proper extension of the real
number field R1/M.
By corolliry 4.1, R114!
Proof.
/.
Theorem 16.
For each P
R1/(PuR1)
= R1,44
Obviously, R2/P will be
is
£
U,
R2/P is an extension of
By theorem 4 and theorem 14,
Proof.
vidsd P
By corollary
R2/N.
13.1, R1/N
i
R2,4.
not maximal in R2.
(R1
a
P)/P Ç
R2/P.
proper extension of R1/1
pro-
However, we are interested in
ths ces. where P is maximal in R2, for then, and only then,
R2/P is
any P
£
First of all, we must determine whether
a
field.
11
is maximal in R2.
In order to do this
we make use
of the
Maximum
rjncip],e.
Let
be a class of sets such that
the union of any linear subclass of Q is an element of
Then Q h
By
.
maximal elements.
linear subclass A of
a
that, if A,B
we mean
then either A
A,
The9rj
17.
Proof.
By the definition of
p L
II
IT,
Q
B
cless A
a
such
A.
or B
contains maximal ideals.
p
p,
::
II
we have
O
implies
II.
Thus we need only to show that there exist maximal elements
in
Let A be any linear subclass of
TI.
U(A)
is an element of H.
Then
a
A, b
e
either A
hence
U(A)
a
B or E
ideal in R2.
also N cU(A).
Since N
Moreover, U(A)
Then a,b
B.
Thus
a
-
b,
p'
A for any A
and U(A)
II.
A,
and
B,
U(A) and
ac
A for each A
we have
A,
N, for if it were, we would
have A su(A) = N contrary to the definition of
I}
Ra.
c
Since A is linear,
A.
Suppose A
ac belong to B.
- b and
is an
,b e U(A) and
For suppose
A.
Then
L1[AZAEA)
B for some A, B
e
il.
II
Finally,
Thus U(A)
so that [i) % U(A).
By the maximum principle
II.
R2
contains maximal
elements.
Thus there exists an ideal
tension field of
R1/.
such that R2/
Moreover, by theorem
homomorphic image of R24\i,
6,
is an
R2/
ex-
is a
29
OTHER RINGS RELATED TO THE REAL NUMBER FIELD
SECTION 4
One might ask whether something similar to what we have
done for R1 and R2 might
riot
ouences of rational numbers,
S be the set of all
be done for other rings of se-
We see
secuences of rational numbers.
immediately that S is
Let
This is indeed possible.
commutative ring with identity, for
a
the sum or product of two secuences (as defined in the in-
troduction) is certainly
a
We shall be concerned,
seouence.
in general, with subrings of S,
ideals in these subrings and
the corresponding nuotint rings.
In order to
make our results meaningful, we should like
to be able to compare the resulting quotient rings with the
real number field
R/L
Thus the following theorem is not
without interest.
Theorem 18.
R
is t)e largest
suing
of S in which N
is an ideal.
Proof,
Let
a}
for an infinite number of subscripts n.
exists
a
Thus
be an unbounded secuence.
monotone, unbounded subsequence
a
O
Moreover, there
[a
}
of [a}. with
k
Let b
O.
b
'k
k
n
(k1,2,3,...) and let
= 1/a
k
(kl,2,3,...).
=
[ab1
Then (be) is
has
a
a
= O for
n
null sequence, but
subsequence converging to
i
and
hence is not null.
In addition, we
may show that there is no proper ideal
in S which contains N,
for, by theorem 7, NS is the smallest
ideal with this property. But NS S, for (l/n} N and
[ni t S. Thus, if fa,.} is any element of S, then [ari =
[i/n}[na1 e NS.
On the other hand, S is not simple, for if R is the set
of all seouences having a finite number of non-zero terms,
is ari ideal in S. By a trivial application of the
maximum principle S must contain maximal ideals. If M is a
maximal ideal in S then S4 is a field and (by theorem 2)
M(\R1 is a prime ideal in R1. Then S/M is an extension
field of the integral domain R1/(MfR1). The latter may or
may not be comparable to the real number field.
A ring which is in many ways more interesting is the
of all seouences [as) such that [(n + l)a -na_1}
set
then
e
R
R1,
Now
I
n+l
that,
sums [i.
so
-kaki]
[(k+l)ak
E
k O
by a well-known theorem on
p. 101],
we
that
have
CC
= a
n
the regularity of Cesaro
R1.
Moreover,
we
have
the following theorem,
is
Iheorem 19.
Proof.
[i}
e
C,
and we need only show
- nan_li
[(n+i)a
identity.
[(n+l).l - n.1} t R1. Thu!
that C1 is a subring of S.
commutative ring with
Firs t: of all, [i}
Let [agi and [bei be
[(n+1)a
a
arbitrary elements of Cr1.
e R1 and
((n+i)b
-nb.1}
e
- na n-1 } - [(n+l)bn - nb n-
Then
R1.
1
=
Thus
31
n(a1 b1
b)
belongs to R1, and hence
1aI
C.
belongs to
[(n+1)a
X
[b}
[a, -
Also
_
na1}[b}
{(n+1)anbrì
na_1b}
R1
and
[a_1)[(n+l)b - nb_1}
y =
belongs to R1.
+
X
{(n+1)a1b
na_1b_1}
Thus
[(n+1)ab
y =
=
* ((n+1)sb
na1b1
+
nanibni}
+
[a1}[b}
t
R1,
Finally, thai
X
[an_i}[bn}
+ y
[abÌ
and hence
[a)[b1
[(n+1)ab -
C.
theorem 2, N1\C1 is
Now by
a
n_ibn'-i}
R1
This completes the proof.
prime ideal In
calling upon the regularity of Cesaro surnmability, we
N(C1 if and only if [a,.) converges to o
see that [a}
na_1) converges, which in turn is possible
and [(n+i)a
if and oniy if [(n+l)ar nani1 converges to O. Thus
Again
1ff [(ri+i)a - n-l1 t N.
is a field.
Theorem 20.
Proof. It is certainly a commutative ring with the
identity, e, which is the eouivalenco class containing [i).
[a1
Lt [a)
e
e Nr%C
C,
and
let
32
if
1.
otherwise.
= O
(rn)
£
Thus
O for n greater than sorno
N, then r
If (a.j
= O,
a
«1
N(\C1,
Let
(b
C.
-nb)
O for any n and
Then b
(l/b).
- tan) -
(l/b1
s
and [(n+l)b
Thus
1b)
[bflØ1),
all belong to R1.
Hence
[i/b)[l/b1)C[b}
+
[b11
n/b1}
= [(n+l)/b
and (l/b}
t
CZ1.
nb1}J
[(n+i)b
£
R1
Now
= [11
-
since
a(1/b)
Since fr}
13y
c
O
O
if
i
otherwise,
a
N(\C1, this completes the proof.
theorems 20 and 4, we have the following theorem.
Theorem 21.
CZ1/(NCCZ1)
is a subfield of the real
number field.
Finally, we remark, thzt the set
of sequences sum-
mable by Cosaro means of the first order is not
It
is,
however, an R-module (as is th
any "proper" method of summability).
a
subring.
summability field of
We might thus construct
33
modules comparable to the real number field (conceived as
module over the rational number field).
However,
case, the theorem corresponding to our lemma
hold,
as
a
i
in this
fails to
so that we would be hard put to redefine the result
ring.
a
1.
2.
:3.
Hardy, G. H.
Press, 1949.
Divergent series.
Oxford, Clarendon
396 p.
Lectures in abstract algebra.
Jacobson, Nathan.
217 p.
voll 1. Princeton, D, Van Nostrand, l9l,
und
Pringsheim, Alfred, Vorlesungen Uber Zahlen
Functionenlebre. Rand 1, Abt, 1. Leipzig, B. G.
Toubner, 1916.
292 p,
4.
Thielman, Henry P. Theory of functions of real
variables. New York, Prentice-Hall, l95. 209 p,
5.
Van der Waerden, B. L. Modern algebra. Vol, 1, Tr. by
258 p.
Fred Blum. Now York, Frederick Ungar, 1949.