Essential extension

Inmathematics,specificallymodule theory,given aringRand anR-moduleMwith a submoduleN,the moduleMis said to be anessential extensionofN(orNis said to be anessential submoduleorlarge submoduleofM) if for every submoduleHofM,

$H\cap N=\{0\}\,$implies that$H=\{0\}\,$

As a special case, anessential left idealofRis aleft idealthat is essential as a submodule of the left module_RR.The left ideal has non-zero intersection with any non-zero left ideal ofR.Analogously, anessential right idealis exactly an essential submodule of the rightRmoduleR_R.

The usual notations for essential extensions include the following two expressions:

$N\subseteq _{e}M\,$(Lam 1999), and$N\trianglelefteq M$(Anderson & Fuller 1992)

Thedualnotion of an essential submodule is that ofsuperfluous submodule(orsmall submodule). A submoduleNis superfluous if for any other submoduleH,

$N+H=M\,$implies that$H=M\,$.

The usual notations for superfluous submodules include:

$N\subseteq _{s}M\,$(Lam 1999), and$N\ll M$(Anderson & Fuller 1992)

Properties

Here are some of the elementary properties of essential extensions, given in the notation introduced above. LetMbe a module, andK,NandHbe submodules ofMwithK$\subseteq$N

• ClearlyMis an essential submodule ofM,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$H\subseteq _{e}M$

UsingZorn's Lemmait is possible to prove another useful fact: For any submoduleNofM,there exists a submoduleCsuch 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 aninjective module.It is then possible to prove that every moduleMhas a maximal essential extensionE(M), called theinjective hullofM.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 containingMcontains a copy ofE(M).

Many properties dualize to superfluous submodules, but not everything. Again letMbe a module, andK,NandHbe submodules ofMwithK$\subseteq$N.

• The zero submodule is always superfluous, and a nonzero moduleMis never superfluous in itself.
• $N\subseteq _{s}M$if and only if$K\subseteq _{s}M$and$N/K\subseteq _{s}M/K$
• $K+H\subseteq _{s}M$if and only if$K\subseteq _{s}M$and$H\subseteq _{s}M$.

Since every module can be mapped via amonomorphismwhose image is essential in an injective module (its injective hull), one might ask if the dual statement is true, i.e. for every moduleM,is there aprojective modulePand anepimorphismfromPontoMwhosekernelis superfluous? (Such aPis called aprojective cover). The answer is "No"in general, and the special class of rings whose right modules all have projective covers is the class of rightperfect rings.

One form ofNakayama's lemmais that J(R)Mis a superfluous submodule ofMwhenMis a finitely-generated module overR.

Generalization

This definition can be generalized to an arbitraryabelian categoryC.Anessential extensionis amonomorphismu:M→Esuch that for every non-zerosubobjects:N→E,thefibre productN×_EM ≠ 0.

In a general category, a morphismf:X→Yis essential if any morphismg:Y→Zis a monomorphism if and only ifg°fis a monomorphism (Porst 1981,Introduction). Takinggto be the identity morphism ofYshows that an essential morphismfmust be a monomorphism.

IfXhas an injective hullY,thenYis the largest essential extension ofX(Porst 1981,Introduction (v)). But the largest essential extension may not be an injective hull. Indeed, in the category of T₁spaces and continuous maps, every object has a unique largest essential extension, but no space with more than one element has an injective hull (Hoffmann 1981).

See also

• Dense submodulesare a special type of essential submodule

References

• Anderson, F.W.; Fuller, K.R. (1992),Rings and Categories of Modules,Graduate Texts in Mathematics,vol. 13 (2nd ed.), Springer-Verlag,ISBN3-540-97845-3
• David Eisenbud,Commutative algebra with a view toward Algebraic GeometryISBN0-387-94269-6
• Hoffmann, Rudolf-E. (1981), "Essential extensions of T₁-spaces ",Canadian Mathematical Bulletin,24(2): 237–240,doi:10.4153/CMB-1981-037-1
• Lam, Tsit-Yuen (1999),Lectures on modules and rings,Graduate Texts in Mathematics No. 189, Berlin, New York:Springer-Verlag,ISBN978-0-387-98428-5,MR1653294
• Mitchell, Barry (1965).Theory of categories.Pure and applied mathematics. Vol. 17. Academic Press.ISBN978-0-124-99250-4.MR0202787.Section III.2
• Porst, Hans-E. (1981), "Characterization of injective envelopes",Cahiers de Topologie et Géométrie Différentielle Catégoriques,22(4): 399–406