Table Of ContentRINGS WITH A POLYNOMIAL IDENTITY
by
LAWRENCE ERNEST BRIDGER
Bachelor of Mathematics, University of Waterloo
Waterloo, Ontario 1968
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in the Department
of
MATHEMATICS.
We accept this thesis as conforming
to the required standard
The University of British Columbia
April 19 70 .
In presenting this thesis in partial fulfilment of the requirements for
an advanced degree at the University of British Columbia, I agree that
the Library shall make it freely available for reference and study.
I further agree tha permission for extensive copying of this thesis
for scholarly purposes may be granted by the Head of my Department or
by his representatives. It is understood that copying or publication
of this thesis for financial gain shall not be allowed without my
written permission.
Department of
The University of British Columbia
Vancouver 8, Canada
Supervisor: Dr. N.J. Divinsky
ABSTRACT.
Since Kaplansky's first paper on the subject of P.I. rings
appeared in 1948, many fruitful results have arisen from the study of
such rings. This thesis attempts to present the most important of these
results in a unified theory.
Chapter I gives the basic notation, definitions, a number of
small lemmas together with Kaplansky's incisive result on primitive
P.I. rings. We investigate also the Kurosh problem for P.I. rings,
providing for such rings an affirmative answer. A rather nice universal
property for P.I. rings which ensures that a ll P.I. rings satisfy some
power of the standard identity is proved.
Chapter II deals with particular types of rings such as rings
without zero divisors and prime rings and culminates in a pair of pretty
results due to Posner and Procesi. We show that prime P.I. rings have
a rather tight structure theory and in fact the restrictions on the
underlying set of coefficients can in this case be relaxed to a very
great extent.
Chapter III is exclusively devoted to P.I. rings with involu
tion. Although such rings are rather specialized much has been
accomplished in this direction in recent years and many beautiful
theorems and proofs have been established, especially by Amitsur and
Martindale.
The source material for chapter IV is primarily Procesi and
i i i.
Amitsur's work on Jacobson rings and Hilbert algebras. Application to
Hilbert's Nullstellensatz and to the Burnside problem are considered.
Finally, chapter V concerns itself completely with generalizat
ions of the preceding four chapters. For the most part these results
do not generalize entirely, but by reducing our demand on polynomial
identities slightly, many remarkably fine results have been proved.
TABLE OF CONTENTS
CHAPTER I
§1: Results from General Ring Theory.
§2: Notation, Definitions, General Results.
§3: The Kurosh Problem.
§4: The Identities of P.I. Rings.
CHAPTER II
§1: A Result of Levitzki and Amitsur.
§2: P.I. Rings with No Nilpotent Ideals.
§3: P.I. Rings with No Zero Divisors.
§4: Prime P.I. Rings.
§5: Prime P.I. Rings (Cont'd).
§6: Two Pretty Results of Posner, Procesi.
§7: Nil and Nilpotence in P.I. Rings.
CHAPTER III \'
§1: Introduction to Rings with Involution.
§2: Preliminary Results. /
§3: Amitsur's Theorem.
§ 4: Other Results.
§5: The Kurosh Problem.
CHAPTER IV
§ 1: A Paper of Amitsur and Procesi.
§2: Extensions of Rings.
§3: Geometric Interpretations.
§4: The Burnside Problem.
CHAPTER V
§1: G.P.I. Rings.
§2: Primitive G.P.I. Rings: Preliminaries.
§3: Primitive G.P.I. Rings.
§4: Prime G.P.I. Rings.
§5: P.M. Rings.
§6: J.P.M. Rings.
§7: G.P.M. Rings.
BIBLIOGRAPHY
vi.
ACKNOWLEDGEMENTS
I wish to thank my advisor Dr. N.J. Divinsky for his suggestion
of the topic and for his time and assistance during the writing of this ;
thesis. I would also like to express my thanks to the National Research
Council of Canada for their financial aid.
\
/
Chapter I
Section 1:
In.this section we merely list a few results from general
ring theory which we do not wish to state or prove in the main body ofj
the thesis. For the most part, these results can be found in many of
the usual books on ring theory (see, for example [16], [20],[ 22], [25],
[26]).
Theorem 1.1: Let A be a primitive ring. There exists a division1!'
ring D such that either A is isomorphic to , or, given any
integer m > 0 , there exists a subring S of A which maps homomorphi
m
cally onto D . ([22], p. 43)
m
Theorem 1.2: Let D be division ring with centre Z and let K be
a maximal subfield of D ; then D ®^ K is a dense ring of linear
transformation on D considered as a vector space over K . ([22].,
p. 95).
Proposition 1.3: If A is a n il semi-simple ring then A[x] , the
ring of polynomials in the commutative indeterminate x over A , is
Jacobson semi-simple. ([4], p.4).
Theorem 1.4: Suppose a ring A satisfies the following four conditions
(11) every direct sum of left ideals contains only a finite number of
non-zero terms.
(lr) every direct sum of right ideals contains only a finite number of
non-zero terms.
(2£) A has A.C.C. on left annihilator ideals.
(2r) A has A.C.C. on right annihilator ideals.
Then A is right quotient simple and left quotient simple, i.e. A has
a full quotient ring A consisting of the set of a ll r x r matrices
over some division ring D ( r is some positive integer).
A (not necessarily commutative) integral domain K is a right
Ore domain if K possesses a classical right quotient field K
Theorem 1.5: Let a ring A be right quotient simple with A = D
Then A . contains a right Ore domain K with K = D , [18]. .
A ring is called right Goldie if it satisfies conditions ',
(lr), (2r) of theorem 1.4. Then we have j
Theorem 1.6: A ring A is a semiprime Goldie, ring i ff it is a right,,
order (i.e. has as its right quotient ring) in a Jacobson semi-simple
algebra Q with D.C.C. A is prime i ff Q is simple. ([22])
By an essential right ideal of a ring A , we shall mean
a right ideal I of . A such that I intersects every non-zero right j
i '
ideal of A in a non-zero fashion. 1
Proposition 1.7: If A is a semiprime right Goldie ring then;
___ - . (a) •, An essential right ideal contains a regular element.
(b) If c is a regular element, cA is essential.
(c) If cx = 0 implies x = 0 then xc = 0 implies x = 0
Finally we state a theorem which is widely used in chapterjilvj,
namely:
Theorem 1.8: A semiprime ideal is the intersection of the prime ideals
containing i t.
Section 2
This section will introduce the notation, basic definitions
and a few elementary properties of rings satisfying a polynomial identity
Let A be an algebra over a field F and let {x^} be a
family (possibly infinite) of noncommuting indeterminates. Denote by
I
F[x ,...,x^] the free algebra over the field F generated by
x ,...,x . Let f(x.,...,x ) e F[x ,...,x ] . If, for every choice
1n m 1 m 1n m
of elements a..,...,a from A we have f(a ,...,a ) = 0 , we say
1
± m 1 m
that A satisfies a polynomial identity over F . In short, A is
a P.I, algebra.
More generally, we might only require that A be an algebra
over a commutative ring F . In some instances we may even be able to
get along fine merely by assuming that A is an ft-algebra for some
domain of operators Q , with the multiplication Q x A -> A satisfying
i
i
some additional conditions ([31], [11]). Usually, the instances in >
which we require only these weaker hypotheses will be clear but those
which are not will be noted. Now, of course, if f = 0 every ring
would satisfy f so that by a P.I. ring, we shall mean a.ring satisfying
a non-zero polynomial identity.
