Table Of ContentTranslations of
MATHEMATICAL
MONOGRAPHS
Volume 87
Ideals of Identities
of Associative Algebras
Aleksandr Robertovich Kemer
American ,Mathematical Society
Ideals of Identities
of Associative Algebras
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 87
Ideals of Identities
of Associative Algebras
Aleksandr Robertovich Kemer
American Mathematical Society
Providence, Rhode Island
AJIEKCAH)P POBEPTOBW1 KEMEP
H)EAJIhI TO)KECTB
ACCOIIHATHBHbIX AJII'EEP
HAYKA», Bapxayn-1988
Translated from the Russian by C. W. Kohls
Translation edited by Ben Silver
1980 Mathematics Subject Classification (1985 Revision). Primary 16A38,
16A03; Secondary 08B20, 08B25, 15A75, 16A42, 16A46.
Library of Congress Cataloging-in-Publication Data
Kemer, Aleksandr Robertovich.
[Idealy tozhdestv assotsiativnykh algebr. English]
Ideals of identities of associative algebras-Aleksandr Robertovich Kemer.
p.; cm.-(Translations of mathematical monographs; v. 87)
Translation of: Idealy tozhdestv assofsiativnykh algebr.
Includes bibliographical references and index.
ISBN 0-8218-4548-9 (alk. paper)
1. Associative algebra. 2. Ideals (Algebra) I. Title. II. Series.
QA251.5.K4613 1991 91-8147
512' .24-dc20 CIP
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1098765432 1 969594939291
Contents
Introduction
1
CHAPTER I. Varieties and Superalgebras 7
§ 1. Technical statements, utilizing the theory of representations
of the symmetric group 7
§2. Grassmann hulls of superalgebras 17
§3. Semiprime varieties. Generalization of the Dubnov-
Ivanov-Nagata-Higman theorem 21
CHAPTER II. Identities of Finitely-Generated Algebras 27
§ 1. Numerical characteristic of T2-ideals 28
§2. A theorem on the decomposition of T2-ideals 31
§ 3. Trace identities 45
§4. Graded identities of finitely-generated superalgebras 55
§5. Solution of Specht's problem 66
§6. On asymptotic bases of identities 68
Bibliography 75
Subject Index 81
V
Introduction
Let F (X) be the free associative algebra over the field F generated by the
,
countable set X, and let A be an arbitrary associative algebra over F. We
say that the algebra A satisfies the polynomial identity f (xl , ... xn) = 0
,
where f(x1, ... , xn) E F(X) and xi E X, if for arbitrary al , ...) , an E A
the equality f(a1, ... , an) = 0 is satisfied in A. An algebra satisfying a
nontrivial polynomial identity (i.e., f(xl , ... , xn) = 0 in F(X)), is called
a PI-algebra.
Commutative algebras, finite-dimensional algebras, and nilpotent algebras
are examples of PI-algebras. A very important example of a PI-algebra is
the Grassmann algebra G, generated by a countable set {el, e2 , ...} with
defining relations ejej = -ejej i, j, = 1, 2, ... (G satisfies the identity of
,
Lie nilpotency of degree 3). The class of PI-algebras is closed with respect to
the formation of subalgebras, homomorphic images, and finite direct sums.
It follows from a theorem of Regev [72] that the class of PI-algebras is also
closed relative to tensor products.
In the survey [28] Jacobson notes that identities were first considered in
connection with configuration theorems in Desarguesian projective geometry
[13]. The next important step in the development of PI-algebras was made by
V. Vagner (see [55]) in proving the commutativity of an ordered PI division
ring and finding some identities of the algebra of matrices of the second
order. Later M. Hall investigated PI division rings with the identity [[x, y] 2
,
z]=0.
A big role in the development of the theory of PI-algebras was played by
a problem posed by Kurosh [46] in 1941: "Is every algebraic algebra locally
finite?" In 1945, Jacobson [26] solved this problem positively for algebraic
algebras of bounded index (these algebras are PI-algebras). In 1946 Levitzki
[54] proved that a finitely-generated PI-algebra over a commutative ring, each
element of which is nilpotent, is nilpotent. Finally, in 1948 Kaplansky solved
Kurosh's problem for PI-algebras over a field [29]. All of these results became
classical and are included in textbooks on ring theory (see [24], [27] and [78]).
The general problem of Kurosh was solved negatively in 1964 by Golod and
Shafarevich [22], [23]. So, with the first results about PI-algebras it became
i
2 INTRODUCTION
clear that the PI condition is a peculiar finiteness condition.
In 1957 A. I. Shirshov proved his remarkable theorem on height.
THEOREM (A. I. Shirshov [75]). For any finitely-generated associative PI-
algebra A over a commutative ring there exist a natural number h and ele-
ments al , ... , an E A such that any element of A can be represented as a
linear combination of elements of the form a"' a,k where k < h
, .
The solution of Kurosh's problem for PI-algebras over a ring follows imme-
diately from this theorem. Shirshov's theorem on height indicates a certain
nearness of finitely-generated PI-algebras to finite-dimensional algebras. The
solution of Kurosh's problem for alternative and special Jordan algebras also
follows from Shirshov's theorem.
A second important problem in the theory of PI-algebras was posed by
Specht ([76]; see also [14]) in 1950: "Does every associative algebra over a
field of characteristic zero have a finite basis of identities?"
The finite basis problem makes sense for algebras over any field, and even
for rings, groups and arbitrary general algebraic systems. The finite basis
problem for groups was solved negatively by Ol'shanskii [64]. For Lie algebras
over a field of finite characteristic this problem was also solved negatively
(Vaughan-Lee [77] and Drenski [15]). In 1973 Kruse [44] and L'vov [56]
proved that every finite ring has a finite basis of identities.
A rather large number of papers have been devoted to Specht's problem
for algebras over a field of characteristic zero. V. N. Latyshev has a big cycle
of papers on this problem [24]-[29]. Many Soviet and Bulgarian algebraists
have worked in this direction (see [18], [20], [33], and [66]). We note the
most important results. In 1978 Latyshev [53] proved that any associative
algebra over a field of characteristic zero satisfying a polynomial identity of
the form
has a finite basis of identities. This result was obtained independently by
Genov [20] and Popov [66]. In 1982 A. V. Yakovlev announced the following
result: The full algebra of matrices of any order over a field of characteristic
zero has a finite basis of identities. The complete positive solution of Specht's
problem was obtained by the author in 1986 [42].
In 1957 Amitsur [2] proved a remarkable theorem: The radical of a finitely-
generated PI-algebra is a nil-ideal. This theorem once again corroborated
that the PI condition is a finiteness condition, and allowed Latyshev at that
time to formulate rather boldly the problem: "Is the radical of a finitely-
generated PI-algebra nilpotent?" (See [ 14]). A big contribution to the solution
of this problem was made by Razmyslov [70]. For algebras over a field of
characteristic zero Latyshev's problem was solved in 1980 by the author [34],
and 1982 Braun [11] solved this problem for algebras over an arbitrary field.
The structure theory of PI-algebras is well developed at the present time.
