Researches On QF Rings

Quasi-Frobenius (QF) rings grew out of the study of Frobenius algebras at the beginning of the last century. The well-known beautiful characterizations of QF rings have attracted many ring experts to do further researches, and some relative books have been published. During the study of QF rings, many challengeable unresolved questions arised, such as Faith-Menal conjecture: every right noetherian and left FP-injective ring is QF. Now the study of QF rings has been a hotspot in international algebraic world.In this dissertation, new characterizations of QF rings are given through strongly Goldie dimension, small injectivity, mininjectivity, simple injectivity and other injectivities associated with some chain conditions. And some new conditions are given for Faith-Menal conjecture to be true. These results improve the results of Wisbauer, Nicholson, Yousif, Osofsky, Chen, Zhou and so on.Strongly Goldie dimension of modules is introduced by the supremum of the Goldie dimensions of quotient modules. Characterizations are given for a module M with SG.dim M=n. At the same time, a new characterization of artinian rings is given by using strongly Goldie dimension. Incidentally, it is proved that if R is right F-injective, then R is semilocal if and only if R is right finitely dimensional, which improves a result of Nicholson and Yousif in 2001: if R is right FP-injective and right Kasch, then R is semilocal if and only if R is right finitely dimensional.Further relations between small injectivity and other injectivities are explored. It is proved that if R is semilocal, then (1) R is right small injective if and only if R is right self-injective,...