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 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 ≠ {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 −1N) ≠ {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−1B for any y in R. Examples If x is