Dense submodule

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Dense submodule. In a right R-module M, a submodule N ⊆ M is called dense if it strengthens the idea of an essential submodule. Intuition: N is large inside M in a way that you can’t avoid hitting N by multiplying elements of M on the right.

A convenient way to describe it: for any y ∈ M, define y−1N = { r ∈ R | y r ∈ N }. This set is always a right ideal of R. Then N is dense in M if and only if, for every nonzero x ∈ M and every y ∈ M, there exists r ∈ R with x r ≠ 0 and y r ∈ N. In words: no nonzero x in M can be killed while moving y into N by right multiplication.

This notion is purely algebraic and does not depend on any topology, although it is related to topological ideas in a broader sense.

Homological view and rational hull. Let E(M) be the injective hull of M—the maximal essential extension of M. There is also a related construction called the rational hull, denoted Ẽ(M), which sits inside E(M). When Ẽ(M) = M, the module M is said to be rationally complete. If R is right nonsingular, then Ẽ(M) = E(M).

Inside E(M), one can describe Ẽ(M using endomorphisms: if S = End_R(E(M)) is the endomorphism ring, then x ∈ E(M) lies in Ẽ(M) exactly when every map in S that sends M to 0 also sends x to 0. Thus some maps in S may be zero on M but nonzero elsewhere, and such x would not be in the rational hull.

Maximal ring of quotients. The maximal right ring of quotients of R relates to dense right ideals of R. It can be described in two standard ways by localizing R at its dense right ideals, producing the largest reasonable ring of right quotients.