In mathematics, specifically module theory, given a ring-module is said to be an essential extension is said to be an essential submodule or large submodule) if for every submodule H,

As a special case, an essential left ideal is a left ideal. Analogously, an essential right ideal.

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,

1. .

The usual notations for superfluous submodules include:

Properties

• , and the zero submodule of a nonzero module is never essential.

Using Zorn's Lemma such that

1. .

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).

.

• is never superfluous in itself.

• .

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. 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 .

See also

• Dense submodules are a special type of essential submodule

References

• David Eisenbud, Commutative algebra with a view toward Algebraic Geometry

• Section III.2