Essential extension

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In mathematics, specifically module theory, given a ring R and R-modules

the module E is an essential extension if for every nonzero submodule

we have

Also, M is then said to be an essential submodule of E.

Some key properties are that given

there exists a maximal submodule

containing M with respect to the property of being an essential extension of M. Given such modules, F being injective implies that E is injective. Finally, given any module M, there is an essential extension E of M that is an injective module, and E is unique up to isomorphism. Such a module E is called the injective envelope of M.

[edit] References

• David Eisenbud, Commutative algebra with a view toward Algebraic Geometry ISBN 0-387-94269-6