# 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 (Johnson 1951), (Utumi 1956) and (Findlay & Lambek 1958).

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 (Storrer 1972) and (Lam 1999, p. 272). 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*.

## Properties

- It can be shown that
*N*is an essential submodule of*M*if and only if for all*y*≠ 0 in*M*, the set*y*⋅(*y*^{−1}*N*) ≠ {0}. Clearly then, every dense submodule is an essential submodule. - If
*M*is a nonsingular module, then*N*is dense in*M*if and only if it is essential in*M*. - A ring is a right nonsingular ring if and only if its essential right ideals are all dense right ideals.
- If
*N*and*N'*are dense submodules of*M*, then so is*N*∩*N'*. - If
*N*is dense and*N*⊆*K*⊆*M*, then*K*is also dense. - If
*B*is a dense right ideal in*R*, then so is*y*^{−1}*B*for any*y*in*R*.

## Examples

- If
*x*is a non-zerodivisor in the center of*R*, then*xR*is a dense right ideal of*R*. - If
*I*is a two-sided ideal of*R*,*I*is dense as a right ideal if and only if the*left*annihilator of*I*is zero, that is, . 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** *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*.

- In one method,
*Ẽ*(*R*) is shown to be module isomorphic to a certain endomorphism ring, and the ring structure is taken across this isomorphism to imbue*Ẽ*(*R*) with a ring structure, that of the maximal right ring of quotients. (Lam 1999, p. 366) - In a second method, the maximal right ring of quotients is identified with a set of equivalence classes of homomorphisms from dense right ideals of
*R*into*R*. The equivalence relation says that two functions are equivalent if they agree on a dense right ideal of*R*. (Lam 1999, p. 370)

## References

- Findlay, G. D.; Lambek, J. (1958), "A generalized ring of quotients. I, II",
*Canadian Mathematical Bulletin*,**1**: 77–85, 155–167, doi:10.4153/CMB-1958-009-3, ISSN 0008-4395, MR 0094370 - Johnson, R. E. (1951), "The extended centralizer of a ring over a module",
*Proc. Amer. Math. Soc.*,**2**: 891–895, doi:10.1090/s0002-9939-1951-0045695-9, ISSN 0002-9939, MR 0045695 - Lam, Tsit-Yuen (1999),
*Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294 - Storrer, Hans H. (1972), "On Goldman's primary decomposition",
*Lectures on rings and modules (Tulane Univ. Ring and Operator Theory)*, Berlin: Springer,**I**(1970–1971): 617–661. Lecture Notes in Math., Vol. 246, doi:10.1007/bfb0059571, MR 0360717 - Utumi, Yuzo (1956), "On quotient rings",
*Osaka Math. J.*,**8**: 1–18, MR 0078966