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 , 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.

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. y^{-1}N = \{r\in R \mid yr\in N \}. 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, x(y^{-1}N) \neq \{0\}. In this case, the relationship is denoted by N\subseteq_d M. Another equivalent definition is homological in nature: N is dense in M if and only if \mathrm{Hom}_R (M/N,E(M)) = \{0\}, where E(M) is the injective hull of M. 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, \ell\cdot \mathrm{Ann}(I)=\{0\}\, . In particular in commutative rings, the dense ideals are precisely the ideals which are faithful modules. 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, \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.

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