A Note on Singular and Nonsingular Modules Relative to Torsion Theories

Let be a hereditary torsion theory. The purpose of this paper is to extend results about singular (resp. nonsingular) modules to -singular (resp. -nonsigular) modules. An R-module is called -singular (resp. -nonsigular) if all its elements (resp. none of its elements except 0) are annihilated by -essential right ideals of R. We proved that, when R is -nonsingular, the quotient of an R-module by its -singular submodule is -nonsingular. Goldie proved that for any submodule N M, the quotient M/N* * is nonsingular. We generalize this result to torsion theoretic setting. Also we introduce the concept of Goldie -closure of a submodule as a generalization of Goldie closure. We proved that it is equivalent to the concept of -essential closure in the case of -nonsingular modules