# 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*,

- implies that

As a special case, an **essential left ideal** of *R* is a left ideal which 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:

- (Lam 1999), and (Anderson & Fuller 1992)

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*,

- implies that .

The usual notations for superfluous submodules include:

- (Lam 1999), and (Anderson & Fuller 1992)

## 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* *N*

- Clearly
*M*is an essential submodule of*M*, 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 *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 with let *M* be a module, and *K*, *N* and *H* be submodules of *M* with *K* *N*.

- 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 (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 which provide their right modules 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.

## See also

- Dense submodules are a special type of essential submodule

## References

- Anderson, F.W.; Fuller, K.R. (1992),
*Rings and Categories of Modules*, Graduate Texts in Mathematics,**13**(2nd ed.), Springer-Verlag, ISBN 3-540-97845-3 - David Eisenbud,
*Commutative algebra with a view toward Algebraic Geometry*ISBN 0-387-94269-6 - Lam, Tsit-Yuen (1999),
*Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 - Mitchell, Barry (1965).
*Theory of categories*. Pure and applied mathematics.**17**. Academic Press. ISBN 978-0-124-99250-4. MR 0202787. Section III.2