Table Of ContentR I N G THEORY
ERNST-AUGUST BEHRENS
Department of Mathematics
McMaster University
Hamilton, Ontario, Canada
TRANSLATED BY
CLIVE REIS
Department of Mathematics
University of Western Ontario
London, Ontario, Canada
1972
ACADEMIC PRESS New York and London
COPYRIGHT 0 1971, BY ACADEMIPCR ESS, INC.
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PRINTED IN THE UNITED STATES OF AMERICA
To Lillie and Jean
Preface
The viability of the theory of commutative rings is evident from
the many papers on the subject which are published each month.
This is not surprising, considering the many problems in algebra and
geometry, and indeed in almost every branch of mathematics, which
lead naturally to the study of various aspects of commutative rings.
In this book we have tried to provide the reader with an introduction
to the basic ideas, results, and techniques of one part of the theory
of commutative rings, namely, multiplicative ideal theory.
The text may be divided roughly into three parts. In the first
part, the basic notions and technical tools are introduced and
developed. In the second part, the two great classes of rings, the
Prufer domains and the Krull rings, are studied in some detail. In
the final part, a number of generalizations are considered. In the
appendix a brief introduction is given to the tertiary decomposition
of ideals of noncommutative rings.
The lengthy bibliography begins with a list of books, some on
commutative rings and others on related subjects. Then follows a
list of papers, all more or less concerned with the subject matter
of the text.
This book has been written for those who have completed a
course in abstract algebra at the graduate level. Preceding the text
there is a discussion of some of the prerequisites which we consider
necessary.
At the end of each chapter are a number of exercises. They are
of three types. Some require the completion of certain technical
details-they might possibly be regarded as busy work. Others
xi
xii PREFACE
contain examples-some of these are messy, but it will be beneficial
for the reader to have some experience with examples. Finally, there
are exercises which enlarge upon some topic of the text or which
contain generalizations of results in the text-the bulk of the exercises
are of this type. A number of exercises are referred to in proofs, and
those proofs cannot be considered to be complete until the relevant
exercises have been done.
We wish to thank those of our colleagues and students who have
commented on our efforts over the years. Special thanks goes to
Thomas Shores for his careful reading of the entire manuscript, and
to our wives for their patience.
Prerequisites
A graduate level course in abstract algebra will provide most of
the background knowledge necessary to read this book. In several
places we have used a little more field theory than might be given
in such a course. The necessary field theory may be found in the
first two chapters of “Algebraic Extensions of Fields by McCarthy,
”
which is listed in the bibliography.
One thing that is certainly required is familiarity with Zorn’s
lemma. Let S be a set. A partial ordering on S is a relation < on
S such that
(i) slsforallsES;
(ii) if s<t and t I s , then s =t; and
< < <
(iii) if s t and t u, then s u.
The set S, together with a partial ordering on S, is called a partially
ordered set.
Let S be a partially ordered set. A subset T of S is called totally
ordered if for all elements s, t E T either s < t or t < s. Let S’ be a
subset of S. An element s E S is called an upper‘bound of S’ if
s‘ Is fo r all s’ E S’. An element s E S is called a maximal element
of S if for an element t ES, s 5 t implies that t = s. Note that S
may have more than one maximal element.
Zorn’s Lemma. Let S be a nonempty partially ordered set. If evuy
totally ordered subset of S has an upper bound in S, then S has a
maximal element.
...
Xlll
xiv PREREQUISITES
If A and B are subsets of some set, then A s B means that A
is a subset of B, and A c B means that A E B but A # B. If S
is a set of subsets of some set, then S is a partially ordered set with
2 as the partial ordering. Whenever we refer to a set of subsets
as a partially ordered set we mean with this partial ordering.
Let S and T be sets and consider a mapping f : S+ T. The
mapping can be described explicitly in terms of elements by writing
sHf(s). If A is a subset of S, we write f(A)= {f(s) \SEA},a nd if
B is a subset of T, we write f-l(B)= {s ISESa nd f(s) EB}.T hus,
f provides us with two mappings, one from the set of subsets of S
into the set of subsets of T and another in the opposite direction.
We assume that the reader can manipulate with these mappings.
If S, T, and U are sets and f : S-t T and g : T+ U are map-
pings, their composition gf : S+ U is defined by (gf) (s) =g(f(s))
for all s E S.
a,,,
On several occasions we shall use the Kronecker delta
which is defined by
1 if i=j,
CHAPTER
Modules
1 RINGS AND MODULES
We begin by recalling the definition of ring. A ring R is a non-
empty set, which we aiso denote by R, together with two binary
+
operations (a,b ) HU b and (a, b) H ab (addition and multiplication,
respectively), subject to the following conditions :
(i) the set R, together with addition, is an Abelian group;
(ii) a(bc) = (ab)c for all a, b, c E R;
+ + + +
(iii) a(b c) = ab ac and (b c)u = ba ca for all a, b, c E R.
Let R be a ring, The identity element of the group of (i) will be
denoted by 0; the inverse of an element a E R considered as an
element of this group will be denoted by -a; a+(-b) will be
written a b. The reader may verify for himself such statements as
-
Oa = a0 = 0 for all a E R,
a( -b) = (-a)b = -(ab) for all a, b E R,
a(b -c) = ab -uc for all a, b, c E R.
A ring R is said to be commutative if ab = ba for all a, b E R.
An element of R is called a unity, and is denoted by 1, if la = a1 = a
for all a E R. If R has a unity, then it has exactly one unity. We shall
assume throughout this book that all rings under consideration have
unities. By a subring of a ring R we mean a ring S such that the
1
2 1 MODULES
set S is a subset of the set R and such that the binary operations of
R yield the binary operations of S when restricted to S x S. By our
assumption concerning unities, both R and S have unities. We shall
consider only those subrings of a ring R which have the same unity
as R.
By a left ideal of a ring R we mean a nonempty subset A of R
such that a - b E A and ra E A for all a, b E A and r E R. By a right
ideal of R we mean a nonempty subset B of R such that a - b E B
and ar E B for all a, b E B and r E R. A left ideal of R which is at the
same time a right ideal of R is czlled an ideal of R. If R is a com-
mutative ring, then the left ideals, right ideals, and ideals of R coincide.
A left ideal, right ideal, or ideal of R is called proper if it is different
from R. The ideal consisting of the element 0 alone is called the zero
ideal of R and will be denoted simply by 0; it is a proper ideal if and
only if R has more than one element.
Let R be a ring. If a E R, then the set
Ra={ra(r ER}
is a left ideal of R, while
aR=(av/r ER}
is a right ideal of R. Since R has a unity, we have a E Ra and a E aR.
The smallest ideal of R containing a, called the principal ideal
generated by a, is the set of elements of R of the form
c
Yi as,
7
where ri , si E R and the sum is finite. This ideal will be denoted by
(a). If R is commutative, Ra = aR = (a).
1.1. Definition. Let R be a ring. A left R-module M is an Abelian
group, written additively, together with a mapping (a, x)t + ax from
R x M into M such that
(1) a(x +y>= ax +by,
+ +
(2) (a b)x = ax bx,
(3) (ab)x = a(bx),
for all a, b E R and x,y E M. A right R-module N is an Abelian
group, written additively, together with a mapping (x,a )t +xa from
N x R into N such that
1 RINGS AND MODULES 3
(1’) (x ++y) a = xu ++y a,
(2’) x(a b) = xu xb,
(3’) x(ab) = (xa)b,
for all a, b E R and x, y E N.
We shall adopt the custom of referring to left R-modules simply
as R-modules. Many of the results concerning modules, left and
right, will be stated for R-modules only. Analogous results hold for
right R-modules.
Let R be a commutative ring and let M be an R-module. If
a E R and x E M, we define xu to be ax. This makes M into a right
R-module. It is immediate that (1’) and (2’) hold. As for (3’), we
note that for all a, b E R and x E M we have x(ab) = (ab)x= (ba)x =
b(ax)= (xa)b. If R is not commutative, then an R-module cannot
necessarily be made into a right R-module in this way.
An R-module M is called unital if lx = x for all x E M. A similar
definition applies to right R-modules. We shall assume throughout
this book that all modules considered, whether left or right modules,
are unital.
An R-module N is called a submodule of an R-module M if N
is a subgroup of M and if the module operation R x N+ N is the
restriction of the module operation R x M-tM to R x N. Suppose
that the nonempty set N is a subset of M. Suppose further that
x - y E N and ax E N for all x, y E N and a E R. Then N, together
with the induced operation of addition, is a subgroup of M. The
mapping (a, x) Ha x from R x N into N makes this subgroup into a
submodule of M. We refer simply to the submodule N.
If L and N are submodules of an R-module M, then their inter-
section L n N is also a submodule of M. More generally, if
{N,I c( E I }i s an arbitrary nonempty family of submodules of M,t hen
n
Na
aEI
+
is a submodule of M. The sum of L and N, denoted by L N, is
defined by
L+N={x+ylx~L and YEN}.
+
It is easily seen that L N is a submodule of M. If N,, . . . , Nka re
submodules of M, then
