ON mu ESSENTIAL AND mu M SINGULAR MODULES

OpenAlex İngilizce

As a generalization of essential submodules Zhou defines a µ-essential submodule pro-vided it has a non-zero intersection with any non-zero submodule in µ for any class µ. Let M be a module. In this article we study δ–essential submodules as a dual of δ-small sub-modules of Zhou where δ = {N ∈ σ[M] : Rej(N,M) = 0} and M = {N ∈ σ[M] : N ¿ N̂}, and also define µ-M–singular modules as modules N ∈ σ[M] such that N ∼ = K/L for some K ∈ σ[M] and L is µ–essential in K. By M–M–singular modules and δ–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.