In mathematics, specifically module theory, given a ringR and R-modules M with a submodule N, the moduleM is said to be an essential extension of N if for every submodule H of M, As a special case, an essential left ideal of R is a left ideal that is essential as a submodule of the left moduleRR. 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 rightR 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. A submodule N is superfluous if for any other submodule H, The usual notations for superfluous submodules include:
Properties
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 KN
Clearly M is an essential submodule of M, and the zero submodule of a nonzero module is never essential.
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 is an injective module. It is then possible to prove that every module M has a maximal essential extension E, called the injective hull of M. The injective hull is necessarily an injective module, and is uniqueup to isomorphism. The injective hull is also minimal in the sense that any other injective module containing M contains a copy of E. Many properties dualize to superfluous submodules, but not everything. Again letM be a module, and K, N and H be submodules of M with KN.
The zero submodule is always superfluous, and a nonzero module M is never superfluous in itself.
if and only if and
if and only if and.
Since every module can be mapped via a monomorphism whose image is essential in an injective module, one might ask if the dual statement is true, i.e. for every module M, is there a projective moduleP and an epimorphism from P onto M whose kernel is superfluous?. The answer is "No" in general, and the special class of rings whose right modules all have projective covers is the class of right perfect rings. One form of Nakayama's lemma is that JM is a superfluous submodule of M when M is a finitely-generated module over R.
Generalization
This definition can be generalized to an arbitraryabelian categoryC. An essential extension is a monomorphism u : M → E such that for every non-zero subobjects : N → E, the fibre productN ×E M ≠ 0.
