Dense submodule

In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If "N" is a dense submodule of "M", it may alternatively be said that ""N" ⊆ "M" is a rational extension". Dense submodules are connected with rings of quotients in noncommutative ring theory. Most of the results appearing here were first established in , and .
It should be noticed that this terminology is different from the notion of a dense subset in general topology. No topology is needed to define a dense submodule, and a dense submodule may or may not be topologically dense in a module with topology.
Definition.
This article modifies exposition appearing in and . Let "R" be a ring, and "M" be a right "R" module with submodule "N". For an element "y" of "M", define
Note that the expression "y"−1 is only formal since it is not meaningful to speak of the module-element "y" being invertible, but the notation helps to suggest that "y"⋅("y"−1"N") ⊆ "N". The set "y" −1"N" is always a right ideal of "R".
A submodule "N" of "M" is said to be a dense submodule if for all "x" and "y" in "M" with "x" ≠ 0, there exists an "r" in "R" such that "xr" ≠ {0} and "yr" is in "N". In other words, using the introduced notation, the set
In this case, the relationship is denoted by
Another equivalent definition is homological in nature: "N" is dense in "M" if and only if
where "E"("M") is the injective hull of "M".
Applications.
Rational hull of a module.
Every right "R" module "M" has a maximal essential extension "E"("M") which is its injective hull. The analogous construction using a maximal dense extension results in the rational hull "Ẽ"("M") which is a submodule of "E"("M"). When a module has no proper rational extension, so that "Ẽ"("M") = "M", the module is said to be rationally complete. If "R" is right nonsingular, then of course "Ẽ"("M") = "E"("M").
The rational hull is readily identified within the injective hull. Let "S"=End"R"("E"("M")) be the endomorphism ring of the injective hull. Then an element "x" of the injective hull is in the rational hull if and only if "x" is sent to zero by all maps in "S" which are zero on "M". In symbols,
In general, there may be maps in "S" which are zero on "M" and yet are nonzero for some "x" not in "M", and such an "x" would not be in the rational hull.
Maximal right ring of quotients.
The maximal right ring of quotients can be described in two ways in connection with dense right ideals of "R".