Locally profinite group

In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring S of a ring R, denoted C_R(S) in this article. It is always the case that C_R(C_R(S)) contains S, and a double centralizer theorem gives conditions on R and S that guarantee that C_R(C_R(S)) is equal to S.

Statements of the theorem

Motivation

The centralizer of a subring S of R given by

${\mathrm{C}}_R(S)=\{r\in R\mid rs=sr\text{ for all }s\in S\}.$

Clearly C_R(C_R(S)) ⊇ S, but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs.

There is another special case of interest. Let M be a right R module and give M the natural left E-module structure, where E is End(M), the ring of endomorphisms of the abelian group M. Every map m_r given by m_r(x) = xr creates an additive endomorphism of M, that is, an element of E. The map r → m_r is a ring homomorphism of R into the ring E, and we denote the image of R inside of E by R_M. It can be checked that the kernel of this canonical map is the annihilator Ann(M_R). Therefore by an isomorphism theorem for rings, R_M is isomorphic to the quotient ring R/Ann(M_R). Clearly when M is a faithful module, R and R_M are isomorphic rings.

So now E is a ring with R_M as a subring, and C_E(R_M) may be formed. By definition one can check that C_E(R_M) = End(M_R), the ring of R module endomorphisms of M. Thus if it occurs that C_E(C_E(R_M)) = R_M, this is the same thing as saying C_E(End(M_R)) = R_M.

Central simple algebras

Perhaps the most common version is the version for central simple algebras, as it appears in Template:Harv:

Theorem: If A is a finite dimensional central simple algebra over a field F and B is a simple subalgebra of A, then C_A(C_A(B)) = B, and moreover the dimensions satisfy

${\mathrm{d}\mathrm{i}\mathrm{m}}_F(B)\cdot {\mathrm{d}\mathrm{i}\mathrm{m}}_F({\mathrm{C}}_A(B))={\mathrm{d}\mathrm{i}\mathrm{m}}_F(A).$

Artinian rings

The following generalized version for Artinian rings (which include finite dimensional algebras) appears in Template:Harv. Given a simple R module U_R, we will borrow notation from the above motivation section including R_U and E=End(U). Additionally, we will write D=End(U_R) for the subring of E consisting of R-homomorphisms. By Schur's lemma, D is a division ring.

Theorem: Let R be a right Artinian ring with a simple right module U_R, and let R_U, D and E be given as in the previous paragraph. Then

$R_U={\mathrm{C}}_E({\mathrm{C}}_E(R_U))$.

Remarks

• In this version, the rings are chosen with the intent of proving the Jacobson density theorem. Notice that it only concludes that a particular subring has the centralizer property, in contrast to the central simple algebra version.
• Since algebras are normally defined over commutative rings, and all the involved rings above may be noncommutative, it's clear that algebras are not necessarily involved.
• If U is additionally a faithful module, so that R is a right primitive ring, then R_U is ring isomorphic to R.

Polynomial identity rings

In Template:Harv, a version is given for polynomial identity rings. The notation Z(R) will be used to denote the center of a ring R.

Theorem: If R is a simple polynomial identity ring, and A is a simple Z(R) subalgebra of R, then C_R(C_R(A)) = A.

Remarks

• This version can be considered to be "between" the central simple algebra version and the Artinian ring version. This is because simple polynomial identity rings are Artinian, but unlike the Artinian version, the conclusion still refers to all central simple subrings of R.

Double centralizer property

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A module M is said to have the double centralizer property or to be a balanced module if C_E(C_E(R_M)) = R_M, where E = End(M) and R_M are as given in the motivation section. In this terminology, the Artinian ring version of the double centralizer theorem states that simple right modules for right Artinian rings are balanced modules.

Notes

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References

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  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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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  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
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  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

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  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
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  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