In the course of the twenties and thirties of the present century, a wide circle of mathematicians became aware of a radical reorganization that had taken place in algebra, one of the oldest branches of mathematics. This reorganization - specifically, the transformation of algebra into a set-theoretical axiomatic science having as its primary object of study the algebraic operations performed on elements of an arbitrary nature-had, of course, been prepared for by the whole preceding development of algebra. It began at the end of the nineteenth century and continued with increasing momentum during the first decades of the twentieth. But it was only the publication, in 1930 and 1931, of the two volumes of van der Waerden's Moderne Algebra that made the ideas, results, and methods of this 'new' algebra accessible to all mathematicians, including non-algebraists.
How great, and sometimes decisive, the impact of this modern algebra was on the development of many domains of mathematics, among which we mention, in the first instance, topology and functional analysis, is common knowledge. At the same time, the last three decades have witnessed an intensive, even stormy development of algebra itself, as well as the discovery of several new links with neighbouring disciplines; as a result, modern algebra - or general algebra, as we prefer to call it - now presents an altogether different appearance from that of thirty years ago.
During these decades, the older branches of general algebra - the theory of fields and of associative and associative-commutative rings, to which van der Waerden's book was mainly devoted - have undergone far-reaching changes. Even more crucial was the reorientation in the theory of groups, the oldest of all the branches of general algebra. At the same time, the theory of rings became more and more a theory of non-associative rings, incorporating as a constituent part the theory of Lie rings and Lie algebras. Topological algebra sprang up and soon occupied a very prominent position, and a parallel development took place in the theory of ordered algebraic structures. The theory of lattices made its appearance and developed rapidly; and the last few years have seen the rise of the parallel theory of categories, which undoubtedly has a most important future. Within the framework of the classical parts of general algebra, independent trends arose: homological algebra, which has already led to numerous results in topology and algebraic geometry; projective algebra, including the elements of projective geometry; and differential algebra, where general algebra yielded direct results in the theory of differential equations. The theory of semigroups and that of quasigroups ceased to be simply theories of 'generalized' groups and found their own paths of development and their own areas of application. Eventually, the general theory of universal algebras came into being, as well as the even more general theory of models, which is interwoven with mathematical logic.
One should have thought that the fundamental ideas and the most important results accumulated in present-day general algebra ought to be part of the scientific equipment of every well-educated mathematician to the same extent as in the thirties, when the majority of candidates in mathematics were examined in modern algebra. However, this is by no means the case: a wide circle of mathematicians today has an acquaintance with the achievements of general algebra that remains rather on the level of the early thirties.
The reason is not difficult to see. The basic textbook from which young Russian mathematicians learn their algebra is still van der Waerden's book. Although this book undoubtedly had a most remarkable and outstanding role in the history of mathematics in the twentieth century, it is now so far removed from the present state of algebra that the author himself changed its title, in the fourth edition, simply to Algebra.
In the non-Russian mathematical literature there are other, more recent books. Some of them, by modernizing the material in van der Waerden's book somewhat, essentially supplement and develop it in the direction of the personal scientific interest of their authors. They are valuable books, but they do not give a true picture of the present state of general algebra. Besides, these books are, as a rule, of considerable bulk and are addressed to algebraists rather than to mathematicians regardless of their specialization. Books of another type present essentially a summary of the most fundamental algebraic concepts and their simplest properties. Although useful as reference books, they do not enable the reader to appreciate the full generality and depth of modern algebraic research: the deepest and most significant results are either omitted entirely or are stated as exercises.
In order to show mathematicians the new look of general algebra, a book of another character is required. Not very large in size, it would have to be addressed to a reader who, having mastered a university course in higher algebra, wishes to extend his algebraic education without necessarily intending to choose algebra as his field of specialization. Of course, this is not to rule out the possibility that even an algebraist, in regard to problems at some remove from his particular interests, might find something useful in the book.
Such a book should not, and could not, replace monographs on the various branches of general algebra. Nor should it merely string together introductory chapters to such monographs. The object of the book would be to exhibit the main branches of modern general algebra, preferably in their mutual interconnection, the exposition being restricted to individual important theorems and aiming straight at these theorems. The selection of a fairly small number of such theorems in each of the main branches of general algebra would inevitably be influenced by the personal judgment of the author of the book. The theorems themselves would by no means have to be given in the greatest generality now attainable.
The contents of the book would, of course, be rather like a mosaic; and, in following the author, the reader would on occasion have to pass, within the confines of a single section, from one branch of general algebra to another. The division of the material into chapters would be so provisory that there could be no question of a scheme of interdependence of the chapters.
At the All-Soviet Congress on Algebra and Number Theory in 1951 (see Usp. Matli. Nauk (1952), vol. 7, part 3, page 167) I spoke of the desirability of a book of this character, and I began to write it in 1956.
During the four years that have since passed, work on the book was repeatedly interrupted and then resumed; several sections were written and rewritten; finished material was rearranged, altered, discarded.... In other words, the work assumed such a character that more and more often I was reminded of Balzac's story Le chef-d'oeuvre inconnu. So it was reasonable to complete the work without getting the book into a form that would correspond to the program laid down above.
The title of the book is entirely justified by the fact that it is based on three extensive special courses I have given at the University of Moscow during the last ten years.
Here and there in the book are statements of certain results that are neither proved nor used. It is hoped that the reader will not skip these passages, which are set apart from the body of the text by the symbol
It goes almost without saying that the inclusion of such supplementary references does not mean that the corresponding sections of the book necessarily represent the most recent results.
When an article in a periodical is quoted in the book, this is rather an incidental matter and should not be taken as material on the history of algebra in the twentieth century. On the other hand, a fairly complete list of books on various branches of general algebra that have appeared in the last thirty years is included in the book. This list contains some survey papers as well.
Inasmuch as the book embodies many plans in its mosaic-like structure, there are very frequent references to preceding material, although in the majority of cases the reader will no doubt find these references superfluous.
I have had the pleasure of presenting the original plan of the book, as well as a number of chapters, some of them in various drafts, before the Seminar on General Algebra at the University of Moscow. To those who participated in the Seminar I offer my sincere thanks for their interest in my work and for their advice and criticism. I also wish to express my warmest thanks to Oleg N Golovin, who has taken on the heavy burden of editing the book, for the great care with which he has read the manuscript and the many helpful suggestions he has made.
Moscow, May1960
A G KUROSH
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