ON μ-ESSENTIAL AND μ-<i>M</i>-SINGULAR MODULES

As a generalization of essential submodules Zhou defines a mu-essential submodule provided it has a non-zero intersection with any non-zero submodule in it for any class p. Let M be a module. In this article we study delta-essential submodules as a dual of delta-small submodules of Zhou where delta = {N is an element of sigma[M] : Rej(N, M) = 0} and M = {N is an element of sigma[M] : N << (N) over cap}, and also define mu-M-singular modules as modules N is an element of sigma[M] such that N congruent to K/L for some K is an element of sigma[M] and L is p-essential in K. By M-M-singular modules and S-M-singular modules a characterization of GCO-modules, and by FC-M-singular modules where FC is the class of finitely cogenerated modules, a characterization of semisimple Artinian rings are given.