Left principal ideal rings
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A. V. Jategaonkar Dept of Mathematics, Cornell University, Ithaca, NY/USA
Left Principal Ideal Rings
Springer-Verlag Berlin· Heidelberg· New York 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
123
A. V. Jategaonkar Dept of Mathematics, Cornell University, Ithaca, NY/USA
Left Principal Ideal Rings
Springer-Verlag Berlin· Heidelberg· New York 1970
This work is subject to copyright. All rights ate reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
Under §.54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © I>y Title No. 3279.
Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 74-114015 Printed iR- Germany.
PREFACE The aim of this monograph is to present a self-contained account of the structure theory and the ideal theory of principal left ideal rings with unity (pli-rings, for short). aspects of pli-rings.
It is not an account of all
Indeed, it contains nothing at all concerning
modules over pli-rings and very little concerning factorization of elements. A glance at the table of contents will reveal a little of what the monograph contains.
Moreover, each chapter has a separate introduction
In each section, the main results are stated as soon as enough definitions are given to make the statements meaningful.
This arrangement
may help to get a detailed idea of the contents unhampered by proofs. A first year graduate course in algebra covering Artinian rings and an acquaintance with ordinals would be enough to read most of the monograph. Some parts of this monograph are taken from my thesis written under the supervision of Professor Newcomb Greenleaf.
I wish to thank
him for the encouragement and advice I received from him.
Thanks are
due to Professor R. E. Johnson who read the first draft of this monograph and made a number of useful suggestion@.
In particular, in §6
of Chapter II, I have followed the simple and elegant method suggested by him.
I wish to thank Professor Alex Rosenberg and Professor P. J.
Hilton for their interest in this monograph.
I wish to thank Mrs.
Manju Bewtra for pointing out some inaccuracies in the first draft.
CONTENTS Preface Chapter I: Left Goldie Rings. ••••••••••••••••••••••••••••••••
1
O.
Terminology and Notation. ••••••.••••••••••••••••••••
2
l.
Left quotient rings" . "
..
5
2.
Goldie's Theorems.
4-4
12
3· 4.
Nil sUbrings . • • • .. " "
"
"
•
"'
"
..
21
....
26
Chapter II: Structure of Pli-rings • • • • • • • • • • • • • • • • • • • • • • • • • • •
34
Small's Theorem. .............................................................. " ..
"
"..
36
.
41
2.
Semi-prime left Goldie semi-pli-rings • • • • • • • •
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