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In the branch of [[abstract algebra]] known as [[ring theory]], a '''left primitive ring''' is a [[Ring (mathematics)|ring]] which has a faithful simple [[left module]]. Well known examples include [[endomorphism ring]]s of [[vector spaces]] and [[Weyl algebra]]s over fields of characteristic zero. | |||
== Definition == | |||
A [[ring (mathematics)|ring]] ''R'' is said to be a '''left primitive ring''' if and only if it has a [[faithful module|faithful]] [[simple module|simple]] [[module (mathematics)|left ''R''-module]]. A '''right primitive ring''' is defined similarly with right ''R''-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in {{harv|Bergman|1964}}. Another example found by Jategaonkar showing the distinction can be found in {{harv|Rowen|1988, p.159}} | |||
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal [[left ideal]] containing no nonzero two-sided [[ideal (ring theory)|ideals]]. The analogous definition for right primitive rings is also valid. | |||
The structure of left primitive rings is completely determined by the [[Jacobson density theorem]]: A ring is left primitive if and only if it is isomorphic to a [[Jacobson density theorem#Topological characterization|dense]] subring of the [[ring of endomorphisms]] of a left [[vector space]] over a division ring. | |||
Another equivalent definition states that a ring is left primitive if and only if it is a [[prime ring]] with a faithful left module of [[length of a module|finite length]]{{Citation needed|date=June 2011}}. | |||
==Properties== | |||
One sided primitive rings are both [[semiprimitive ring]]s and [[prime ring]]s. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive. | |||
For a left [[Artinian ring]], it is known that the conditions "left primitive", "right primitive", "prime", and "[[simple ring|simple]]" are all equivalent, and in this case it is a [[semisimple ring]] isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive"="right primitive"="prime". | |||
A [[commutative ring]] is left primitive if and only if it is a [[field (mathematics)|field]]. | |||
Being left primitive is a [[Morita equivalence|Morita invariant property]]. | |||
== Examples == | |||
Every [[simple ring]] ''R'' with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that [[maximal ideal#Properties|''R'' has a maximal left ideal]] ''M'', and the fact that the [[quotient module]] ''R''/''M'' is a simple left ''R''-module, and that its [[annihilator (ring theory)|annihilator]] is a proper two-sided ideal in ''R''. Since ''R'' is a simple ring, this annihilator is {0} and therefore ''R''/''M'' is a faithful left ''R''-module. | |||
[[Weyl algebra]]s over fields with [[characteristic (algebra)|characteristic]] zero are primitive, and since they are [[domain (ring theory)|domain]]s, they are examples without minimal one-sided ideals.<!-- how about positive char? --> | |||
===Full linear rings=== | |||
A special case of primitive rings is that of ''full linear rings''. A '''left full linear ring''' is the ring of ''all'' linear transformations of an infinite dimensional left vector space over a division ring. (A '''right full linear ring''' differs by using a right vector space instead.) In symbols, <math>R=\mathrm{End}(_D V)\,</math> where ''V'' is a vector space over a division ring ''D''. It is known that ''R'' is a left full linear ring if and only if ''R'' is [[von Neumann regular]], [[Injective module#Self-injective rings|left self-injective]] with [[socle (mathematics)|socle]] soc(<sub>''R''</sub>''R'')≠{0}. {{harv|Goodearl|1991|p=100}} Through linear algebra arguments, it can be shown that <math>\mathrm{End}(_D V)\,</math> is isomorphic to the ring of [[matrix ring#Examples|row finite matrices]] <math>\mathbb{RFM}_I(D)\,</math>, where ''I'' is an index set whose size is the dimension of ''V'' over ''D''. Likewise right full linear rings can be realized as column finite matrices over ''D''. | |||
Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring ''R'' is always left primitive. When dim<sub>''D''</sub>''V'' is finite ''R'' is a square matrix ring over ''D'', but when dim<sub>''D''</sub>''V'' is infinite, the set of finite rank linear transformations is a proper two-sided ideal of ''R'', and hence ''R'' is not simple. | |||
== References == | |||
*{{Citation | doi=10.1090/S0002-9939-1964-0167497-4 | last1=Bergman | first1=G. M. | title=A ring primitive on the right but not on the left | jstor=2034527 | mr=0167497 | year=1964 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=15 | pages=473–475 | issue=3 | publisher=American Mathematical Society}} [http://www.jstor.org/stable/2034929 p. 1000 errata] | |||
*{{citation | |||
|author=Goodearl, K. R. | |||
|title=von Neumann regular rings | |||
|edition=2 | |||
|publisher=Robert E. Krieger Publishing Co. Inc. | |||
|place=Malabar, FL | |||
|date=1991 | |||
|pages=xviii+412 | |||
|isbn=0-89464-632-X | |||
|mr=1150975 (93m:16006)} | |||
}} | |||
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 | mr=1838439 | year=2001}} | |||
*{{citation | |||
|author=Rowen, Louis H. | |||
|title=Ring theory. Vol. I | |||
|series=Pure and Applied Mathematics | |||
|volume=127 | |||
|publisher=Academic Press Inc. | |||
|place=Boston, MA | |||
|year=1988 | |||
|pages=xxiv+538 | |||
|isbn=0-12-599841-4 | |||
|mr=940245 (89h:16001) | |||
}} | |||
[[Category:Ring theory]] | |||
[[Category:Algebraic structures]] | |||
Latest revision as of 19:47, 26 September 2013
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
Definition
A ring R is said to be a left primitive ring if and only if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in Template:Harv. Another example found by Jategaonkar showing the distinction can be found in Template:Harv
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring.
Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite lengthPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park..
Properties
One sided primitive rings are both semiprimitive rings and prime rings. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive"="right primitive"="prime".
A commutative ring is left primitive if and only if it is a field.
Being left primitive is a Morita invariant property.
Examples
Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields with characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.
Full linear rings
A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle soc(RR)≠{0}. Template:Harv Through linear algebra arguments, it can be shown that is isomorphic to the ring of row finite matrices , where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D.
Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.
References
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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.
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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
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