Dense submodule
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
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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
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In this case, the relationship is denoted by
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Another equivalent definition is homological in nature: "N" is dense in "M" if and only if
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where "E"("M") is the injective hull of "M". 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,
formula_6
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. The maximal right ring of quotients can be described in two ways in connection with dense right ideals of "R".
Applications
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.