Dense submodule explained

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

y-1N=\{r\inR\midyr\inN\}

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−1N) ⊆ N. The set y −1N 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 ≠  and yr is in N. In other words, using the introduced notation, the set

x(y-1N)\{0\}

In this case, the relationship is denoted by

N\subseteqdM

Another equivalent definition is homological in nature: N is dense in M if and only if

HomR(M/N,E(M))=\{0\}

where E(M) is the injective hull of M.

Properties

Examples

\ellAnn(I)=\{0\}

. In particular in commutative rings, the dense ideals are precisely the ideals which are faithful modules.

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 (M) which is a submodule of E(M). When a module has no proper rational extension, so that (M) = M, the module is said to be rationally complete. If R is right nonsingular, then of course (M) = E(M).

The rational hull is readily identified within the injective hull. Let S=EndR(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,

\tilde{E}(M)=\{x\inE(M)\mid\forallf\inS,f(M)=0\impliesf(x)=0\}

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

See main article: article.

The maximal right ring of quotients can be described in two ways in connection with dense right ideals of R.