ABSTRACT
In this chapter, the authors discuss the study of extending modules with various chain conditions, like ACC or DCC for annihilators or essential submodules. This will yield a number of characterizations of quasi-Frobenius rings and structure theorems for more general classes of rings. The chain conditions, the authors investigate include the ACC and DCC on left annihilators, the ACC and DCC on essential left ideals, and other chain conditions related to these. For formulating the first theorem on QF-rings we need the concept of dual modules. The authors consider conditions on annihilators under which the singular ideal must be nilpotent. They prove some results about extending modules satisfying certain chain conditions on essential submodules. The authors consider some relationships between hereditary noetherian rings and the restricted minimum condition and characterize some classes of rings with chain conditions.
