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, 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'', :

H\cap N=\\,

implies that

H=\\,

As a special case, an essential left ideal of ''R'' is a left ideal that 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, an 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: :

N\subseteq_e M\,

, and

N\trianglelefteq M

The

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule ''N'' is superfluous if for any other submodule ''H'', :

N+H=M\,

implies that

H=M\,

. The usual notations for superfluous submodules include: :

N\subseteq_s M\,

, and

N\ll M

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''

\subseteq

''N'' *Clearly ''M'' is an essential submodule of ''M'', and the zero submodule of a nonzero module is never essential. *

K\subseteq_e M

if and only if

K\subseteq_e N

and

N\subseteq_e M

K \cap H \subseteq_e M

if and only if

K\subseteq_e M

and

H\subseteq_e M

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 :

N\oplus C \subseteq_e M

. 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 in