Essential extension


In mathematics, specifically module theory, given a ring and an -module with a submodule, the module is said to be an essential extension of if for every submodule H of,
As a special case, an essential left ideal of is a left ideal that is essential as a submodule of the left module. The left ideal has non-zero intersection with any non-zero left ideal of. Analogously, an essential right ideal is exactly an essential submodule of the right module.
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 is superfluous if for any other submodule,
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 be a module, and, and be submodules of with
  • Clearly is an essential submodule of, and the zero submodule of a nonzero module is never essential.
  • if and only if and
  • if and only if and
Using Zorn's Lemma it is possible to prove another useful fact: For any submodule of, there exists a submodule 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 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.
Many properties dualize to superfluous submodules, but not everything. Again let be a module, and, and be submodules of with.
  • The zero submodule is always superfluous, and a nonzero module 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 module P 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 arbitrary abelian category. An essential extension is a monomorphism such that for every non-zero subobject, the fibre product.
In a general category, a morphism f : X → Y is essential if any morphism g : Y → Z is a monomorphism if and only if g ° f is a monomorphism. Taking g to be the identity morphism of Y shows that an essential morphism f must be a monomorphism.
If X has an injective hull Y, then Y is the largest essential extension of X. But the largest essential extension may not be an injective hull. Indeed, in the category of T1 spaces and continuous maps, every object has a unique largest essential extension, but no space with more than one element has an injective hull.