A Note on Singular and Nonsingular Modules Relative to Torsion Theories

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