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In [[mathematics]], especially [[ring theory]], a '''regular ideal''' can refer to multiple concepts.


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In [[operator theory]], a right [[ideal (ring theory)|ideal]] <math>\mathfrak{i}</math> in a (possibly) [[non-unital ring]] ''A'' is said to be '''regular''' (or '''modular''') if there exists an element ''e'' in ''A'' such that <math>ex - x \in \mathfrak{i}</math> for every <math>x \in A</math>. {{harv|Jacobson|1956}}


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In [[commutative algebra]] a '''regular ideal''' refers to an ideal containing a non-[[zero divisor]].<ref>Non-zerodivisors in commutative rings are called ''regular elements''.</ref> {{harv|Larsen|McCarthy|1971|loc=p.42}} This article will use "regular element ideal" to help distinguish this type of ideal.


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A two-sided ideal <math>\mathfrak{i}</math> of a ring ''R'' can also be called a (von Neumann) '''regular ideal''' if for each element ''x'' of <math>\mathfrak{i}</math> there exists a ''y'' in <math>\mathfrak{i}</math> such that ''xyx''=''x''. {{harv|Goodearl|1991|loc=p.2}} {{harv|Kaplansky|1969|loc=p.112}}


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Finally, '''regular ideal''' has been used to refer to an ideal ''J'' of a ring ''R'' such that the [[quotient ring]] ''R''/''J'' is [[von Neumann regular ring]].<ref>Burton, D.M. (1970) ``A first course in rings and ideals.'' Addison-Wesley. Reading, Massachusetts .</ref> This article will use "quotient von Neumann regular" to refer to this type of regular ideal.


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Since the adjective ''regular'' has been overloaded, this article adopts the alternative adjectives ''modular'', ''regular element'' ''von Neumann regular'', and ''quotient von Neumann regular'' to distinguish between concepts.


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==Properties and examples==


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===Modular ideals===
The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
 
A two-sided ideal <math>\mathfrak{i}</math> is modular if and only if <math>A/\mathfrak{i}</math> is unital. In a unital ring, every ideal is modular since choosing ''e''=1 works for any right ideal. So, the notion is more interesting for non-unital rings such as [[Banach algebra]]s. From the definition it is easy to see that an ideal containing a modular ideal is itself modular.
 
Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained in a maximal right ideal.{{sfn|Jacobson|1956|loc=p.6}} The intersection of all maximal right ideals which are modular is the [[Jacobson radical]].{{sfn|Kaplansky|1948|loc=Lemma 1}}
 
;Examples
* In the non-unital ring of even integers, (6) is regular (<math>e = 4</math>) while (4) is not.
* Let ''M'' be a simple right A-module. If ''x'' is a nonzero element in ''M'', then the annihilator of ''x'' is a regular maximal right ideal in ''A''.
* If ''A'' is a ring without maximal right ideals, then ''A'' cannot have even a single modular right ideal.
 
===Regular element ideals===
Every ring with unity has at least one regular element ideal: the trivial ideal ''R'' itself. Regular element ideals of commutative rings are [[essential submodule|essential ideals]]. In a [[semiprime ring|semiprime]] right [[Goldie ring]], the converse holds: essential ideals are all regular element ideals. {{harv|Lam|1999|loc=p.342}}
 
Since the product of two ''regular elements'' (=non-zerodivisors) of a commutative ring ''R'' is again a regular element, it is apparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal.
 
;Examples
* In an [[integral domain]], every nonzero element is a regular element, and so every nonzero ideal is a regular element ideal.
* The [[nilradical of a ring|nilradical]] of a commutative ring is composed entirely of [[nilpotent element]]s, and therefore no element can be regular. This gives an example of an ideal which is not a regular element ideal.
* In an [[Artinian ring]], each element is either [[invertible]] or a zero divisor. Because of this, such a ring only has one regular element ideal: just ''R''.
 
===Von Neumann regular ideals===
From the definition, it is clear that ''R'' is a [[von Neumann regular ring]] if and only if ''R'' is a von Neumann regular ideal. The following statement is a relevant lemma for von Neumann regular ideals:
 
'''Lemma''': For a ring ''R'' and proper ideal ''J'' containing an element ''a'', there exists and element ''y'' in ''J'' such that ''a''=''aya'' if and only if there exists an element ''r'' in ''R'' such that ''a''=''ara''. '''Proof''': The "only if" direction is a tautology. For the "if" direction, we have ''a''=''ara''=''arara''. Since ''a'' is in ''J'', so is ''rar'', and so by setting ''y''=''rar'' we have the conclusion.
 
As a consequence of this lemma, it is apparent that every ideal of a von Neumann regular ring is a von Neumann regular ideal. Another consequence is that if ''J'' and ''K'' are two ideals of ''R'' such that ''J''⊆''K'' and ''K'' is a von Neumann regular ideal, then ''J'' is also a von Neumann regular ideal.
 
If ''J'' and ''K'' are two ideals of ''R'', then ''K'' is von Neumann regular if and only if both ''J'' is a von Neumann regular ideal and ''K''/''J'' is a von Neumann regular ring.{{sfn|Goodearl|1991|loc=p.2}}
 
Every ring has at least one von Neumann regular ideal, namely {0}. Furthermore, every ring has a maximal von Neumann regular ideal containing all other von Neumann regular ideals, and this ideal is given by
:<math>M=\{x\in R\mid RxR\text{ is a von Neumann regular ideal }\}</math>.
 
;Examples
* As noted above, every ideal of a von Neumann regular ring is a von Neumann regular ideal.
* It is well known that a [[local ring]] which is also a von Neumann regular ring is a [[division ring]]. Let ''R'' Be a local ring which is ''not'' a division ring, and denote the unique maximal right ideal by ''J''. Then ''R'' cannot be von Neumann regular, but ''R''/''J'', being a division ring, is a von Neumann regular ring. Consequently, ''J'' cannot be a von Neumann regular ideal, even though it is maximal.
* A [[simple ring|simple]] [[domain (ring theory)|domain]] which is not a division ring has the minimum possible number of von Neumann regular ideals: only the {0} ideal.
 
===Quotient von Neumann regular ideals===
If ''J'' and ''K'' are quotient von Neumann regular ideals, then so is ''J''∩''K''.
 
If ''J''⊆''K'' are proper ideals of ''R'' and ''J'' is quotient von Neumann regular, then so is ''K''. This is because quotients of ''R''/''J'' are all von Neumann regular rings, and an  [[Isomorphism theorems#Third isomorphism theorem 2|isomorphism theorem]] for rings establishing that ''R''/''K''≅(''R''/''J'')/(''J''/''K''). In particular if ''A'' is ''any'' ideal in ''R'' the ideal ''A''+''J'' is quotient von Neumann regular if ''J'' is.
 
;Examples
* Every proper ideal of a von Neumann regular ring is quotient von Neumann regular.
* Any maximal ideal in a commutative ring is a quotient von Neumann regular ideal since ''R''/''M'' is a field. This is not true in general because for noncommutative rings ''R''/''M'' may only be a simple ring, and may not be von Neumann regular.
* Let ''R'' be a local ring which is not a division ring, and with maximal right ideal ''M'' . Then ''M'' is a quotient von Neumann regular ideal, since ''R''/''M'' is a division ring, but ''R'' is not a von Neumann regular ring.
* More generally in any [[semilocal ring]] the [[Jacobson radical]] ''J'' is quotient von Neumann regular, since ''R''/''J'' is a [[semisimple ring]], hence a von Neumann regular ring.
 
== References ==
{{Reflist}}
 
*{{citation  |author=Goodearl, K. R.  |title=von Neumann regular rings  |edition=2  |publisher=Robert E. Krieger Publishing Co. Inc.  |place=Malabar, FL  |year=1991  |pages=xviii+412  |isbn=0-89464-632-X  |mr=1150975 }}
 
*{{citation|author=Jacobson, Nathan |title=Structure of rings |series=American Mathematical Society, Colloquium Publications, vol. 37 |publisher=American Mathematical Society|place=190 Hope Street, Prov., R. I. |year=1956 |pages=vii+263 |mr=0081264}}
 
*{{citation |last=Kaplansky|first=Irving |title=Dual rings |journal=Ann. of Math. (2) |volume=49 |year=1948 |pages=689–701 |issn=0003-486X  |mr=0025452}}
 
* {{cite book
| author = Irving Kaplansky
| year = 1969
| title = Fields and Rings
| publisher = The University of Chicago Press
}}
 
*{{citation|author1=Larsen, Max. D. |author2=McCarthy, Paul J. |title=Multiplicative theory of ideals |note=Pure and Applied Mathematics, Vol. 43 |publisher=Academic Press |place=New York |year=1971 |pages=xiv+298 |mr=0414528}}
 
*{{Springer
|title = Modular ideal
|id = m/m064450.htm
|first =K.A.
|last=Zhevlakov}}
 
{{DEFAULTSORT:Regular Ideal}}
[[Category:Ring theory]]
[[Category:Ideals]]

Revision as of 20:31, 3 November 2013

In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.

In operator theory, a right ideal in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that for every . Template:Harv

In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor.[1] Template:Harv This article will use "regular element ideal" to help distinguish this type of ideal.

A two-sided ideal of a ring R can also be called a (von Neumann) regular ideal if for each element x of there exists a y in such that xyx=x. Template:Harv Template:Harv

Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring.[2] This article will use "quotient von Neumann regular" to refer to this type of regular ideal.

Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular, regular element von Neumann regular, and quotient von Neumann regular to distinguish between concepts.

Properties and examples

Modular ideals

The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.

A two-sided ideal is modular if and only if is unital. In a unital ring, every ideal is modular since choosing e=1 works for any right ideal. So, the notion is more interesting for non-unital rings such as Banach algebras. From the definition it is easy to see that an ideal containing a modular ideal is itself modular.

Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained in a maximal right ideal.Template:Sfn The intersection of all maximal right ideals which are modular is the Jacobson radical.Template:Sfn

Examples
  • In the non-unital ring of even integers, (6) is regular () while (4) is not.
  • Let M be a simple right A-module. If x is a nonzero element in M, then the annihilator of x is a regular maximal right ideal in A.
  • If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal.

Regular element ideals

Every ring with unity has at least one regular element ideal: the trivial ideal R itself. Regular element ideals of commutative rings are essential ideals. In a semiprime right Goldie ring, the converse holds: essential ideals are all regular element ideals. Template:Harv

Since the product of two regular elements (=non-zerodivisors) of a commutative ring R is again a regular element, it is apparent that the product of two regular element ideals is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal.

Examples
  • In an integral domain, every nonzero element is a regular element, and so every nonzero ideal is a regular element ideal.
  • The nilradical of a commutative ring is composed entirely of nilpotent elements, and therefore no element can be regular. This gives an example of an ideal which is not a regular element ideal.
  • In an Artinian ring, each element is either invertible or a zero divisor. Because of this, such a ring only has one regular element ideal: just R.

Von Neumann regular ideals

From the definition, it is clear that R is a von Neumann regular ring if and only if R is a von Neumann regular ideal. The following statement is a relevant lemma for von Neumann regular ideals:

Lemma: For a ring R and proper ideal J containing an element a, there exists and element y in J such that a=aya if and only if there exists an element r in R such that a=ara. Proof: The "only if" direction is a tautology. For the "if" direction, we have a=ara=arara. Since a is in J, so is rar, and so by setting y=rar we have the conclusion.

As a consequence of this lemma, it is apparent that every ideal of a von Neumann regular ring is a von Neumann regular ideal. Another consequence is that if J and K are two ideals of R such that J⊆K and K is a von Neumann regular ideal, then J is also a von Neumann regular ideal.

If J and K are two ideals of R, then K is von Neumann regular if and only if both J is a von Neumann regular ideal and K/J is a von Neumann regular ring.Template:Sfn

Every ring has at least one von Neumann regular ideal, namely {0}. Furthermore, every ring has a maximal von Neumann regular ideal containing all other von Neumann regular ideals, and this ideal is given by

.
Examples
  • As noted above, every ideal of a von Neumann regular ring is a von Neumann regular ideal.
  • It is well known that a local ring which is also a von Neumann regular ring is a division ring. Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J. Then R cannot be von Neumann regular, but R/J, being a division ring, is a von Neumann regular ring. Consequently, J cannot be a von Neumann regular ideal, even though it is maximal.
  • A simple domain which is not a division ring has the minimum possible number of von Neumann regular ideals: only the {0} ideal.

Quotient von Neumann regular ideals

If J and K are quotient von Neumann regular ideals, then so is J∩K.

If J⊆K are proper ideals of R and J is quotient von Neumann regular, then so is K. This is because quotients of R/J are all von Neumann regular rings, and an isomorphism theorem for rings establishing that R/K≅(R/J)/(J/K). In particular if A is any ideal in R the ideal A+J is quotient von Neumann regular if J is.

Examples
  • Every proper ideal of a von Neumann regular ring is quotient von Neumann regular.
  • Any maximal ideal in a commutative ring is a quotient von Neumann regular ideal since R/M is a field. This is not true in general because for noncommutative rings R/M may only be a simple ring, and may not be von Neumann regular.
  • Let R be a local ring which is not a division ring, and with maximal right ideal M . Then M is a quotient von Neumann regular ideal, since R/M is a division ring, but R is not a von Neumann regular ring.
  • More generally in any semilocal ring the Jacobson radical J is quotient von Neumann regular, since R/J is a semisimple ring, hence a von Neumann regular ring.

References

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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
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

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  1. ↑ Non-zerodivisors in commutative rings are called regular elements.
  2. ↑ Burton, D.M. (1970) ``A first course in rings and ideals. Addison-Wesley. Reading, Massachusetts .
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