Essential extension - Citizendia

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

$M\subseteq E$ ,

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

$N\subseteq E$ ,

we have

$N\cap M\ne 0$ . Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

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

Some key properties are that given

$M\subseteq F$ ,

there exists a maximal submodule

$E\subseteq F$

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. In Mathematics, especially in the area of Abstract algebra known as Module theory, an injective module is a module Q that shares certain 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Such a module E is called the injective envelope of M. In Mathematics, especially in the area of Abstract algebra known as Module theory, the injective hull (or injective envelope) of a module is

References

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

David Eisenbud (born 8 April, 1947) is an American Mathematician.

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