Essential extension

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

:$Msubseteq\; E$,

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

:$Nsubseteq\; E$,

we have

:$Ncap\; M\; e\; 0$.

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

Some key properties are that given

:$Msubseteq\; F$,

there exists a maximal submodule

:$Esubseteq\; F$

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.

References

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