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Dense submodule

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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

y − 1 N = { r ∈ R ∣ y r ∈ N } {\displaystyle y^{-1}N=\{r\in R\mid yr\in N\}\,}

Note that the expression y 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⋅(yN) ⊆ N. The set y 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

x ( y − 1 N ) ≠ { 0 } {\displaystyle x(y^{-1}N)\neq \{0\}\,}

In this case, the relationship is denoted by

N ⊆ d M {\displaystyle N\subseteq _{d}M\,}

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

H o m R ( M / N , E ( M ) ) = { 0 } {\displaystyle \mathrm {Hom} _{R}(M/N,E(M))=\{0\}\,}

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 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 yB 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, ℓ ⋅ A n n ( I ) = { 0 } {\displaystyle \ell \cdot \mathrm {Ann} (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 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=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,

E ~ ( M ) = { x ∈ E ( M ) ∣ ∀ f ∈ S , f ( M ) = 0 ⟹ f ( x ) = 0 } {\displaystyle {\tilde {E}}(M)=\{x\in E(M)\mid \forall f\in S,f(M)=0\implies f(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

Main article: Maximal 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

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