Kuratowski's closure-complement problem

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

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

As a special case, an essential left ideal of R is a left ideal which is essential as a submodule of the left module _RR. 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_R

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

$N{\subseteq }_{e}M$ Template:Harv, and $N⊴M$ Template:Harv

The 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,

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

The usual notations for superfluous submodules include:

$N{\subseteq }_{s}M$ Template:Harv, and $N\ll M$ Template:Harv

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 $\subset$ 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 $N{\subseteq }_{e}M$
• $H\cap K{\subseteq }_{e}M$ if and only if $K{\subseteq }_{e}M$ and $N{\subseteq }_{e}M$

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 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).

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.

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

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 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 rings.

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 ×_E M ≠ 0.

See also

• Dense submodules are a special type of essential submodule

References

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• David Eisenbud, Commutative algebra with a view toward Algebraic Geometry ISBN 0-387-94269-6
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Template:Mitchell TOC Section III.2