vili PREFACE however, assuming the existence of a unity is a convenience, and so we have built it into the definition. We have also required in the definition of a module that the unity of the ring act as an identity on the module. Whatever notational scheme one adopts, it is important that students learn of the existence of common alternatives how else could they read the literature or attend courses from other lecturers? For this reason, I have attempted to mention competing notations and definitions whenever appropriate in the text. At the end of each chapter, there is a fairly extensive list of problems. Few of these are routine exercises, and some of them I consider to be quite difficult. The purpose of these problems is not just to give practice with the definitions and with understanding the theorems. My hope is that by working these problems, students will get a feeling of what it is actually like to do algebra, and not just to learn it. (I should mention that when I teach my algebra course, I assign five problems per week.) This is not a "scholarly" book I have not attempted to trace back to their sources the various definitions, lemmas, theorems, and ideas presented here. I have credited items to individuals in cases such as the "Sylow theorems," the "Jacobson radical," and the "Hilbert basis theorem" where such attribution is standard and well known and in other situations where it seemed appropriate. Even in these cases, however, I have not given bibliographic references to the original sources. I am grateful to the following reviewers for their helpful comments: Michael Aschbacher, California Institute of Technology Carl Widland, Indiana University Edward Green, Virginia Polytechnic Institute and State University Seth Warner, Duke University E. Graham Evans, Jr., University of Illinois, Urbana-Champaign Robert L. Griess, Jr., University of Michigan, Ann Arbor Ancel Mewborn, Univer- sity of North Carolina at Chapel Hill Peter Norman, University of Massachusetts, Amherst and Gerald Janusz, University of Illinois, Urbana-Champaign. Let me close this preface by expressing my hope that this book will engender, in some readers at least, the same excitement and love for algebra that I received from Professor Loomis in myfirstyear of graduate school. /. Martin Isaacs Madison, WI
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