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 í 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.
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机译：让我们将其作为遗传扭转理论。本文的目的是将关于奇异（分别为非奇异）模块的结果扩展到t异同（分别为t-非奇异）模块。如果R模块的所有元素（除了0以外，其他任何元素）都没有被R的t本质右理想所消灭，则称其为t奇异（分别为t非正整数）。我们证明了，当R为t-如果是非奇数，则R模块的t奇数子模块的商为t非奇数。 Goldie证明，对于任何子模块NíM，商M / N **是非奇数的。我们将这个结果推广到扭转理论上。我们还介绍了子模块的Goldie t闭包概念，作为Goldie闭包的概括。我们证明了，在t非奇异模块的情况下，它等效于t本质封闭的概念。关键字：扭转理论，扭转模块，无扭转模块，t密度子模块，（非）奇异模块。
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