Euler–Heisenberg Lagrangian: Difference between revisions
Jump to navigation
Jump to search
en>Dicklyon dash fix |
en>Yobot m →References: WP:CHECKWIKI error fixes / special characters in sortkey fixed using AWB (9427) |
||
| Line 1: | Line 1: | ||
In [[mathematics]], the notion of a '''divisor''' originally arose within the context of arithmetic of whole numbers. See the article on [[divisor]]s for this simplest example. With the development of abstract [[Ring (mathematics)|rings]], of which the [[integer]]s are the [[archetype]], the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of [[commutative ring]]s because of its relationship with the [[ideal (ring theory)|ideal]] structure of such rings. | |||
==Definition== | |||
Let ''R'' be a commutative ring,<ref>In this article, rings are assumed to have a 1.</ref> and let ''a'' and ''b'' be elements of ''R''. | |||
One says that ''a'' '''divides''' ''b'' in ''R'' if there exists an element ''x'' in ''R'' with <math> ax=b </math>.<ref>Bourbaki, p. 97</ref> | |||
In this case, one writes <math> a \mid b </math>, and one says also that ''a'' is a '''divisor''' of ''b'', or that ''b'' is a '''[[multiple (mathematics)|multiple]]''' of ''a''. | |||
Notes: These definitions make sense in any commutative [[semigroup]] ''R'', but they are used primarily when this semigroup is the multiplicative [[monoid]] of a commutative ring. Divisors are useful also in non-commutative contexts; one must then discuss [[right divisor|left]] and [[right divisor]]s. | |||
Elements ''a'' and ''b'' of a commutative ring are '''associates''' if both <math> a \mid b </math> and <math> b \mid a </math>. The associate relationship is an [[equivalence relation]] on ''R'', and hence divides ''R'' into [[disjoint sets|disjoint]] [[equivalence class]]es. | |||
== Properties == | |||
Statements about divisibility can be translated into statements about [[principal ideal]]s. For instance, | |||
* One has <math> a \mid b </math> if and only if <math> (b) \subseteq (a) </math>. | |||
* Elements ''a'' and ''b'' are associates if and only if <math> (a) = (b) </math>. | |||
* An element ''u'' is a unit if and only if ''u'' is a divisor of every element of ''R''. | |||
* An element ''u'' is a unit if and only if <math> (u) = R </math>. | |||
* If <math> a = b u </math> for some unit ''u'', then ''a'' and ''b'' are associates. If ''R'' is an [[integral domain]], then the converse is true. | |||
== Zero as a divisor, and zero divisors == | |||
* Some authors require ''a'' to be nonzero in the definition of divisor, but this causes some of the properties above to fail. | |||
* If one interprets the definition of divisor literally, every ''a'' is a divisor of 0, since one can take ''x''=0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element ''a'' in a commutative ring a [[zero divisor]] if there exists a ''nonzero'' ''x'' such that ''ax''=0.<ref>Bourbaki, p. 98</ref> | |||
== Notes == | |||
{{reflist}} | |||
== References == | |||
* {{citation |author= N. Bourbaki |title=Algebra I, Chapters 1–3 |publisher=Springer-Verlag |year=1989}}. | |||
{{Citizendium|Divisor (ring theory)}} | |||
== See also == | |||
* [[Divisor]] | |||
* [[Zero divisor]] | |||
[[Category:Ring theory]] | |||
Revision as of 10:13, 20 August 2013
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. See the article on divisors for this simplest example. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.
Definition
Let R be a commutative ring,[1] and let a and b be elements of R. One says that a divides b in R if there exists an element x in R with .[2] In this case, one writes , and one says also that a is a divisor of b, or that b is a multiple of a.
Notes: These definitions make sense in any commutative semigroup R, but they are used primarily when this semigroup is the multiplicative monoid of a commutative ring. Divisors are useful also in non-commutative contexts; one must then discuss left and right divisors.
Elements a and b of a commutative ring are associates if both and . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes.
Properties
Statements about divisibility can be translated into statements about principal ideals. For instance,
- One has if and only if .
- Elements a and b are associates if and only if .
- An element u is a unit if and only if u is a divisor of every element of R.
- An element u is a unit if and only if .
- If for some unit u, then a and b are associates. If R is an integral domain, then the converse is true.
Zero as a divisor, and zero divisors
- Some authors require a to be nonzero in the definition of divisor, but this causes some of the properties above to fail.
- If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x=0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax=0.[3]
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- 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.