This page intentionally left blank QUASI-FROBENIUSRINGS Thestudyofquasi-Frobeniusringsgrewoutofthetheoryofgrouprepresen- tationsinthe1940sandhasproducedanenormousbodyofresults.Thisbook makes no attempt to be encyclopedic but provides an elementary account of the basic facts about these rings at a level allowing researchers and graduate studentstogainentrytothefield.Manyearlierresultsaboutself-injectiverings areextendedtothemuchwiderclassofmininjectiverings;themethodsused unify and simplify what is known in the area and so bring the reader up to current research. Sufficient background knowledge can be found in standard texts on noncommutative rings. However, appendices on Morita equivalence; onperfect,semiperfect,andsemiregularrings;andontheCamps–Dicksthe- oremareincludedtomakethebookself-contained.Afterthebasicresultsare established in Chapters 1 through 6, recent work is reviewed on three open problemsinthefield(theFaithconjecture,theFGF-conjecture,andtheFaith– Menal conjecture). Some new results are provided and new and old methods forattackingtheseproblemsareoutlinedinaneasilyaccessibleformat. W.K.NicholsonisProfessorofMathematicsattheUniversityofCalgary. M.F.YousifisProfessorofMathematicsatTheOhioStateUniversity. CAMBRIDGETRACTSINMATHEMATICS GeneralEditors B.BOLLOBA´S,W.FULTON,A.KATOK,F.KIRWAN, P.SARNAK 158 Quasi-Frobenius Rings QUASI-FROBENIUS RINGS W.K.NICHOLSON UniversityofCalgary M.F.YOUSIF TheOhioStateUniversity    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521815932 © W. K. Nicholson & M. F. Yousif 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 -  isbn-13 978-0-511-06510-1 eBook (NetLibrary) -  isbn-10 0-511-06510-8 eBook (NetLibrary) -  isbn-13 978-0-521-81593-2 hardback -  isbn-10 0-521-81593-2 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. ForKathleenandEman