Essential extension
In mathematics, specifically module theory, given a ring R and R-modules M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for every submodule H of M,
As a special case, an essential left ideal of R is a left ideal which is essential as a submodule of the left module RR. The left ideal has non-zero intersection with any non-zero left ideal of R. Analogously, and essential right ideal is exactly an essential submodule of the right R module RR
The usual notations for essential extensions include the following two expressions:
The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule N is superfluous if for any other submodule H,
The usual notations for superfluous submodules include:
Here are some of the elementary properties of essential extensions, given in the notation introduced above. Let M be a module, and K, N and H be submodules of M with K N
Using Zorn's Lemma it is possible to prove another useful fact: For any submodule N of M, there exists a submodule C such that
Furthermore, a module with no proper essential extension (that is, if the module is essential in another module, then it is equal to that module) is an injective module. It is then possible to prove that every module M has a maximal essential extension E(M), called the injective hull of M. The injective hull is necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injective module containing M contains a copy of E(M).
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