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 that is essential as a submodule of the left module "R""R". 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 "R""R".
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:
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 "K" formula_9 "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").
Many properties dualize to superfluous submodules, but not everything. Again let "M" be a module, and "K", "N" and "H" be submodules of "M" with "K" formula_17 "N".
Since every module can be mapped via a monomorphism whose image is essential in an injective module (its injective hull), 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? (Such a "P" is called a projective cover). 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 J("R")"M" 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 C. An essential extension is a monomorphism "u" : "M" → "E" such that for every non-zero subobject "s" : "N" → "E", the fibre product "N" ×"E" M ≠ 0.