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Concept
Essential extension
Summary
In mathematics, specifically module theory, given a ring R and an R-module 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 that is essential as a submodule of the left module RR. The left ideal has non-zero intersection with any non-zero left ideal of R. Analogously, an essential right ideal is exactly an essential submodule of the right R module RR.
The usual notations for essential extensions include the following two expressions:
and
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:
and
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 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.
Official source
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