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| In [[mathematics]], specifically [[module theory]], given a [[ring (mathematics)|ring]] ''R'' and ''R''-[[module (mathematics)|module]]s ''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'',
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| :<math>H\cap N=\{0\}\,</math> implies that <math>H=\{0\}\,</math>
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| As a special case, an '''essential left ideal''' of ''R'' is a left ideal which is essential as a submodule of the left module <sub>''R''</sub>''R''. The left ideal has non-zero intersection with any non-zero left ideal of ''R''. Analogously, and '''essential right ideal''' is exactly an essential submodule of the right ''R'' module ''R''<sub>''R''</sub>
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| The usual notations for essential extensions include the following two expressions:
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| :<math>N\subseteq_e M\,</math> {{harv|Lam|1999}}, and <math>N\trianglelefteq M</math> {{harv|Anderson|Fuller|1992}}
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| The [[duality (mathematics)|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'',
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| :<math>N+H=M\,</math> implies that <math>H=M\,</math>.
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| The usual notations for superfluous submodules include:
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| :<math>N\subseteq_s M\,</math> {{harv|Lam|1999}}, and <math>N\ll M</math> {{harv|Anderson|Fuller|1992}}
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| ==Properties==
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| 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'' <math> \subset</math> ''N''
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| *Clearly ''M'' is an essential submodule of ''M'', and the zero submodule of a nonzero module is never essential.
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| *<math>K\subseteq_e M</math> if and only if <math>K\subseteq_e N</math> and <math>N\subseteq_e M</math>
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| *<math>H\cap K\subseteq_e M</math> if and only if <math>K\subseteq_e M</math> and <math>N\subseteq_e M</math>
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| Using [[Zorn's Lemma]] it is possible to prove another useful fact:
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| For any submodule ''N'' of ''M'', there exists a submodule ''C'' such that
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| :<math>N\oplus C \subseteq_e M</math>. | |
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| 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'').
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| Many properties dualize to superfluous submodules, but not everything. Again with let ''M'' be a module, and ''K'', ''N'' and ''H'' be submodules of ''M'' with ''K'' subset ''N''.
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| *The zero submodule is always superfluous, and a nonzero module ''M'' is never superfluous in itself.
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| *<math>N\subseteq_s M</math> if and only if <math>K\subseteq_s M</math> and <math>N/K \subseteq_s M/K</math>
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| *<math>H+K\subseteq_s M</math> if and only if <math>N\subseteq_s M</math> and <math>K\subseteq_s M</math>.
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| Since every module can be mapped via a [[monomorphism]] whose image is essential in an injective module (its injective hull), one might ask if the dual statement is true, i.e. for every module ''M'', is there a [[projective module]] ''P'' and an [[epimorphism]] from ''P'' onto ''M'' whose [[kernel (algebra)|kernel]] is superfluous? (Such a ''P'' is called a [[projective cover]]). The answer is "''No''" in general, and the special class of rings which provide their right modules projective covers is the class of right [[perfect ring]]s.
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| ==Generalization== | |
| This definition can be generalized to an arbitrary [[abelian category]] '''C'''. An '''essential extension''' is a [[monomorphism]] ''u'' : ''M'' → ''E'' such that for every non-zero [[subobject]] ''s'' : ''N'' → ''E'', the [[fibre product]] ''N'' ×<sub>''E''</sub> M ≠ 0.
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| ==See also==
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| *[[Dense submodule]]s are a special type of essential submodule
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| ==References==
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| *{{citation |last=Anderson |first=F.W. |coauthor=K.R. Fuller |title=Rings and Categories of Modules | publisher=Graduate Texts in Mathematics, Vol. 13, 2 nd Ed., Springer-Verlag |year=1992 |ISBN= 0-387-97845-3 | ISBN= 3-540-97845-3}}
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| * [[David Eisenbud]], ''Commutative algebra with a view toward Algebraic Geometry'' ISBN 0-387-94269-6
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| * {{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
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| * {{Mitchell TOC}} Section III.2
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| [[Category:Abstract algebra]]
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| [[Category:Commutative algebra]]
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| [[Category:Module theory]]
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