Singular submodule

In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) "R" module "M" has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in "R". In set notation it is usually denoted as formula_1. For general rings, formula_2 is a good generalization of the torsion submodule tors("M") which is most often defined for domains. In the case that "R" is a commutative domain, formula_3.
If "R" is any ring, formula_4 is defined considering "R" as a right module, and in this case formula_4 is a twosided ideal of "R" called the right singular ideal of "R". Similarly the left handed analogue formula_6 is defined. It is possible for formula_7.
Definitions.
Here are several definitions used when studying singular submodule and singular ideals. 
In the following, "M" is an "R" module:
In rings with unity it is always the case that formula_11, and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules.
Properties.
Some general properties of the singular submodule include:
Examples.
Right nonsingular rings are a very broad class, including reduced rings, right (semi)hereditary rings, von Neumann regular rings, domains, semisimple rings, Baer rings and right Rickart rings.
For commutative rings, being nonsingular is equivalent to being a reduced ring.
Important theorems.
Johnson's Theorem (due to R. E. Johnson ) contains several important equivalences. For any ring "R", the following are equivalent:
Right nonsingularity has a strong interaction with right self injective rings as well.
Theorem: If "R" is a right self injective ring, then the following conditions on "R" are equivalent: right nonsingular, von Neumann regular, right semihereditary, right Rickart, Baer, semiprimitive. 
The paper used nonsingular modules to characterize the class of rings whose maximal right ring of quotients have a certain structure.
Theorem: If "R" is a ring, then formula_21 is a right full linear ring if and only if "R" has a nonsingular, faithful, uniform module. Moreover, formula_21 is a finite direct product of full linear rings if and only if "R" has a nonsingular, faithful module with finite uniform dimension.