Fuzzy Maximal Sub-Modules

The first part of this paper investigates the structure of δ-local modules. We prove that the following statements are equivalent for a module M: (i) M is δ-local; (ii) M is a coatomic module with either (a) M is a semisimple module having a maximal submodule N such that N is projective and M/N is singular, or (b) M has a unique essential maximal submodule K ≤ M such that for every maximal submodule L ≠ K, M/L is projective. The second part establishes some properties of finitely generated amply δ-supplemented modules.

Lifting Morita equivalence to categories of fuzzy modules

A module M is called virtually semisimple (resp. virtually extending) if every submodule (resp. complement submodule) of M is isomorphic to a direct summand of M. It is known that virtually extending modules is a generalization of virtually semisimple modules. In this paper, the relationships between virtually extending modules and other generalizations of virtually semisimple modules are examined. Moreover, we introduce a new generalization of virtually semisimple modules; namely CH modules: We say a module M is a c-epi-retractable (or briefly CH module) if any complement submodule of M is a homomorphic image of M. CH modules contains the class of virtually extending modules and the class of epi-retractable modules. We also give some basic properties of this new module class.

In this chapter, we shall study the central theme of this book, namely, representation of a finite group G over a field K. We establish some of the basic properties using the theory of semi-simple modules over the so called “group algebra” of G over K. (We shall illustrate these results for several important special cases of groups such as S n (Chapter 5), A n (Chapter 6), B n (Chapter 7) and D n (Chapter 8) below.)