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