Quasi-Frobenius Rings and Generalizations QF-3 and QF-1 Rings Notes
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351
Hiroyuki Tachikawa Tokyo University of Education, Tokyo/Japan
Ouasi-Frobenius Rings and Generalizations OF-3 and OF-1 Rings Notes by Claus Michael Ringel
Springer-Verlag Berlin· Heidelberg· New York 1973
Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
351
Hiroyuki Tachikawa Tokyo University of Education, Tokyo/Japan
Ouasi-Frobenius Rings and Generalizations OF-3 and OF-1 Rings Notes by Claus Michael Ringel
Springer-Verlag Berlin· Heidelberg· New York 1973
AMS Subject Classifications (1970): 16A36
ISBN 3-540-06501-6 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-06501-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are 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. © by Springer-Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-14490. Printed in Germany. Offsetdruck: Julius Beltz, Hernsbach/Bergstr.
INTRODUCTION
The class of quasi-Frobenius rings (short:
QF rings)
is one of the most interesting classes of non-semisimple rings.
Since R. M. Thrall [53J introduced QF-l, QF-2 and
QF-3 rings as its generalizations, already a quarter of a century has passed.
During these years important
developments have taken place in the theory of these rings such as the appearance of homological dimensions, perfect and semi-perfect rings, Morita duality and maximal quotient rings.
The purpose of these lectures was to
give an up-to-date account of these developments. Though there are many generalizations of QF-3 algebras to arbitrary rings, we shall adopt in the following only Thrall's original definition:
a ring R is said to be
left QF-3 provided there exists a (unique) minimal faithful left R-module RU,
That is, RU is faithful, and is
isomorphic to a direct summand of every faithful left R-module.
Here, it is to be noted that we do not impose
any other restriction on R.
Then, as was proved by
Colby-Rutter [9J, RU is projective and isomorphic to the injective hull of a direct sum of finitely many simple left R-modules.
We can identify RU with a left ideal Re,
= e.
Right QF-3 rings are defined similarly, and rings
IV which are both left QF-3 and right QF-3 will simply be called QF-3 rings. Assume now that R is a QF-3 ring and minimal faithful modules with e 2
and fR
r
e, f2
e
are R Then we
n.
may consider fRf and eRe as rings, and fRe as an fRf-eRebimodule.
The first remarkable fact is that the functors
Homf Rf (- , fRe) and Hom ( - , fRe) define a Morita duality e He between the category of reflexive left fRf-modules and the category of reflexive right eRe-modules, and, moreover, fRffR and Re e Re bot
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