Commalg - User contributions [en]

Urysohn space

2026-05-22T01:42:28Z

Vipul:

==Definition==

A topological space <math>X</math> is termed a '''Urysohn space''' if given any two distinct points <math>x,y \in X</math>, there exists a continuous function <math>f:X \to [0,1]</math> such that <math>f(x) = 0</math>, <math>f(y) = 1</math>.

==Facts==

* [[Natural map from topological space to max-spectrum of ring of continuous real-valued functions is an injection iff the space is Urysohn]]

==External links==

===Primary subject wiki link===

* [[Topospaces:Urysohn space]]

MediaWiki:Sitenotice

2024-09-12T04:59:47Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-09-12T04:59:29Z

Vipul:

MediaWiki:Sitenotice

2024-09-08T17:57:24Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.'''<br/>
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-08-11T17:25:22Z

Vipul:

Commalg:429 Too Many Requests error

2024-08-11T17:15:36Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

File:Site search autocompletion working.png

2024-08-11T17:14:20Z

Vipul:

File:Site search autocompletion broken.png

2024-08-11T17:13:07Z

Vipul:

Commalg:Enabling site search autocompletion

2024-08-11T17:11:52Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Commalg). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in t..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Commalg).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

MediaWiki:Sitenotice

2024-08-11T17:10:36Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-08-11T17:05:14Z

Vipul:

User:Vipul/Sandbox

2024-08-11T17:01:09Z

Vipul:

MediaWiki:Sitenotice

2024-08-11T16:53:32Z

Vipul:

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

User:Vipul

2023-12-08T05:33:33Z

Vipul:

<section begin="math-formula-on-user-vipul-page"/>
math formula tests removed

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>3! * 6 = 36</math>
* <math>(3 + 4)^3 = 343</math>
* <math>5^{1 + 2} = 125</math>
* <math>6^{2 + 1} = 216</math>
* <math>(3 + 4)^3 = 343</math>
* <math>2^{8 - 1} = 128</math>

<section end="math-formula-on-user-vipul-page"/>

User:Vipul

2023-12-08T05:33:25Z

Vipul:

User:Vipul

2023-11-18T23:42:49Z

Vipul:

User:Vipul

2023-11-14T17:31:26Z

Vipul:

User:Vipul

2023-11-12T21:42:11Z

Vipul:

User:Vipul

2023-11-12T19:36:44Z

Vipul:

User:Vipul

2023-11-12T16:32:05Z

Vipul:

User:Vipul

2023-11-12T16:14:52Z

Vipul:

User:Vipul

2023-11-12T16:14:40Z

Vipul:

<section begin="math-formula-on-user-vipul-page"/>
math formula tests removed

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>3! * 6 = 36</math>

<section end="math-formula-on-user-vipul-page"/>

{{#ask: [[Broader than::Membrane transport protein]][[Narrower than::Membrane protein]]|format = list|limit = 10}}

Euclidean domain

2023-11-12T16:11:29Z

Vipul: /* Definition */

{{integral domain property}}

==Definition==

===Symbol-free definition===

An [[integral domain]] is said to be '''Euclidean''' if it admits a [[defining ingredient::Euclidean norm]].

===Definition with symbols===

An [[integral domain]] <math>R</math> is termed a '''Euclidean domain''' if there exists a function <math>N</math> from the set of nonzero elements of <math>R</math> to the set of nonnegative integers satisfying the following properties:

* <math>N(x) = 0</math> if and only if <math>x</math> is a unit
* Given nonzero <math>a</math> and <math>b</math> in <math>R</math>, there exist <math>q</math> and <math>r</math> such that <math>a = qb + r</math> and either <math>r = 0</math> or <math>N(r) < N(b)</math>.

We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.

Such a function <math>N</math> is called a [[Euclidean norm]] on <math>R</math>.

===Caveats===

* The definition of Euclidean norm does not require the ring to be an integral domain. A commutative unital ring that admits a Euclidean norm is termed a Euclidean ring.
* The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].

==Examples==

===Standard examples===

* The [[ring of rational integers]] <math>\mathbb{Z}</math> is a Euclidean domain with Euclidean norm defined by the absolute value. {{proofat|[[Ring of integers is Euclidean with norm equal to absolute value]]}}
* The [[polynomial ring over a field]] <math>k[x]</math> is a Euclidean domain with Euclidean norm defined by the degree of a polynomial. This is, in fact a ''uniquely'' Euclidean norm. and hence the polynomial ring over a field is a uniquely Euclidean domain. {{proofat|[[Polynomial ring over a field is uniquely Euclidean with norm equal to degree]]}}

===Other examples===

* The [[ring of Gaussian integers]] <math>\mathbb{Z}[i]</math> is a Euclidean domain with Euclidean norm equal to the norm in the sense of a quadratic integer ring. {{proofat|[[Ring of Gaussian integers is norm-Euclidean]]}}
* A quadratic integer ring, or more generally, a [[ring of integers in a number field]], is termed [[norm-Euclidean ring of integers in a number field]] if it is Euclidean with respect to the algebraic norm. Since there is a correspondence between number fields and their rings of integers, we often abuse language and say that the number field itself is norm-Euclidean.
* Any [[discrete valuation ring]] is a Euclidean domain where the norm of an element is given by the largest power of the irreducible that divides it. For instance, the [[formal power series ring over a field]] is a Euclidean domain, where the norm of a formal power series is the smallest <math>n</math> for which the coefficient of <math>x^n</math> that is nonzero.

===Pathological examples===

On a field, ''any'' norm function is Euclidean. This is because we can always choose a quotient so that the remainder is zero.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::uniquely Euclidean domain]] || there is a [[Euclidean norm]] for which Euclidean division is unique. || || || {{intermediate notions short|Euclidean domain|uniquely Euclidean domain}}
|-
| [[Weaker than::Polynomial ring over a field]] || it can be written as the [[polynomial ring]] <math>K[x]</math> for a [[field]] <math>K</math>. || || || {{intermediate notions short|Euclidean domain|polynomial ring over a field}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::multi-stage Euclidean domain]] || || || || {{intermediate notions short|multi-stage Euclidean domain|Euclidean domain}}
|-
| [[Stronger than::principal ideal domain]] || [[integral domain]] that is a [[principal ideal ring]] || [[Euclidean implies PID]] || [[PID not implies Euclidean]] || {{intermediate notions short|principal ideal domain|Euclidean domain}}
|-
| [[Stronger than::Bezout domain]] || [[integral domain]] in which every [[finitely generated ideal]] is [[principal ideal|principal]] || || || {{intermediate notions short|Bezout domain|Euclidean domain}}
|-
| [[Stronger than::unique factorization domain]] || || || || {{intermediate notions short|unique factorization domain|Euclidean domain}}
|-
| [[Stronger than::Dedekind domain]] || Noetherian, normal, one-dimensional domain || || || {{intermediate notions short|Dedekind domain|Euclidean domain}}
|-
| [[Stronger than::Noetherian domain]] || [[integral domain]] and every ideal is finitely generated || || || {{intermediate notions short|Noetherian domain|Euclidean domain}}
|-
| [[Stronger than::Noetherian ring]] || every ideal is finitely generated || || || {{intermediate notions short|Noetherian ring|Euclidean domain}}
|}

===Properties of Euclidean norms===

Euclidean norms can in general be very weirdly behaved, but some Euclidean norms are good. For a complete list of properties of Euclidean norms (i.e., properties against which a given Euclidean norm can be tested), refer:

[[:Category:Properties of Euclidean norms]]

Here are some important properties that most ''typical'' Euclidean norms satisfy:

* [[Multiplicatively monotone Euclidean norm]]

==Metaproperties==

{{not poly-closed curing property}}

The polynomial ring over a Euclidean domain need not be a Euclidean domain. One example is the polynomial ring with integer coefficients, which is not a Euclidean domain; another example is the polynomial ring in ''two'' variables over a field (which can be viewed as the polynomial ring in one variable, over the polynomial ring over a field).

Template:Property implication (generic)

2023-11-12T16:08:04Z

Vipul:

{{quotation|This article gives the statement and possibly, proof, of an implication relation between two [[{{{context space}}} property|{{{context space}}} properties]]. That is, it states that every {{{context space}}} satisfying the first {{{context space}}} property {{#if:{{{stronger|}}}|(i.e., [[Fact about::{{{stronger}}}]])}} must also satisfy the second {{{context space}}} property {{#if:{{{weaker|}}}|(i.e., [[Fact about::{{{weaker}}}]])}}<br>[[:Category:{{{context space}}} property implications|View all {{{context space}}} property implications]] <nowiki>|</nowiki> [[:Category:{{{context space}}} property non-implications|View all {{{context space}}} property non-implications]] <nowiki>|</nowiki>[[Help:{{{context space}}} property implication lookup|Get help on looking up {{{context space}}} property implications/non-implications]]<br>{{#if:{{{stronger|}}}|[[Special:SearchByProperty/Fact about/{{{stronger}}}|Get more facts about {{{stronger}}}]]}}<nowiki>|</nowiki>{{#if:{{{weaker|}}}|[[Special:SearchByProperty/Fact about/{{{weaker}}}|Get more facts about {{{weaker}}}]]}}}}<includeonly>{{#ifeq:{{NAMESPACE}}|{{ns:0}}|[[Category: {{{context space}}} property implications]][[Page class::Fact| ]]}}</includeonly><noinclude>[[Category: Property implication specification templates]]</noinclude>

Euclidean implies principal ideal

2023-11-12T16:06:37Z

Vipul:

{{curing property implication|stronger = Euclidean ring|weaker = principal ideal ring}}

==Statement==

===Property-theoretic statement===

The [[property of commutative unital rings]] of being a [[Euclidean ring]] is stronger than the property of being a [[principal ideal ring]].

===Verbal statement===

Any [[Euclidean ring]] is a [[principal ideal ring]]. In particular, any [[Euclidean domain]] is a [[principal ideal domain]].

==Proof==

The key idea behind the proof is to, given an ideal, exhibit an element that generates the ideal. The claim is as follows:

''Consider the [[Euclidean norm]] as a function on the ideal. Pick any element with minimum norm (such an element exists because the set of possible norms is a well-ordered set). Then, the ideal is generated by that element''.

(Note that when the ideal is the zero ideal, the element of smallest possible norm is zero, whose norm is assumed to be <math>\infty</math>).

Let's prove this claim. Suppose <math>R</math> is a [[Euclidean ring]] with [[Euclidean norm]] <math>N</math>, and <math>I</math> is an [[ideal]] inside <math>R</math>. Let <math>d</math> be the smallest possible norm among elements of <math>I</math>, and let <math>a \in I</math> be an element such that <math>N(a) = d</math>. Now pick and <math>c \in I</math>. We want to show that <math>c</math> is a multiple of <math>a</math>.

By the division algorithm over <math>R</math>, there exist <math>q,r \in R</math> such that:

<math>c = aq + r, \qquad N(r) < N(a)</math> or <math>r = 0</math>

Now note that <math>r = c - aq</math>, and since <math>c,a \in I</math> and <math>I</math> is an [[ideal]], <math>r \in I</math>. Since <math>a</math> was chosen with minimum norm, <math>N(r) < N(a)</math> is impossible, so <math>r = 0</math>, and thus <math>c</math> is a multiple of <math>a</math>.

User:Vipul

2023-11-12T16:04:04Z

Vipul:

User:Vipul

2023-11-12T05:36:49Z

Vipul:

<section begin="math-formula-on-user-vipul-page"/>
math formula tests removed

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>(2\sqrt{4})! = 24</math>

<section end="math-formula-on-user-vipul-page"/>

User:Vipul/Lst

2023-11-12T05:10:14Z

Vipul: Created page with "{{#lst:User:Vipul|math-formula-on-user-vipul-page}}"

User:Vipul

2023-11-12T05:09:36Z

Vipul:

<section begin="math-formula-on-user-vipul-page"/>
math formula tests removed

<math>9^{\sqrt{7 + 2}} = 729</math>
<section end="math-formula-on-user-vipul-page"/>

User:Vipul

2023-11-12T04:33:36Z

Vipul:

math formula tests removed

<math>9^{\sqrt{7 + 2}} = 729</math>

User:Vipul

2023-11-12T04:02:48Z

Vipul:

math formula tests removed

User:Vipul

2023-11-12T00:11:13Z

Vipul:

math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>5^{2 + 1} = 125</math>
* <math>36 = 3! * 6</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^2 = 25</math>

User:Vipul

2023-11-12T00:10:54Z

Vipul:

I am Vipul Naik, a B.Sc. (Hons) Math student at [http://www.cmi.ac.in Chennai Mathematical Institute]. This wiki (on Commutative Algebra) is my brainchild, and I've modelled it on my {{gpwiki}}.

Other wikis of interest are:

* {{dfwiki}}
* {{ncwiki}}

Learn more about me at my [http://www.cmi.ac.in/~vipul Home Page].

math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>5^{2 + 1} = 125</math>
* <math>36 = 3! * 6</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^2 = 25</math>

User:Vipul

2023-11-11T22:16:01Z

Vipul:

User:Vipul

2023-11-11T22:12:46Z

Vipul:

User:Vipul

2023-11-11T22:07:30Z

Vipul:

User:Vipul

2023-11-11T22:06:55Z

Vipul:

MediaWiki:Sidebar

2023-11-11T22:06:11Z

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User:Vipul

2023-11-11T22:00:17Z

Vipul:

Commalg:Privacy policy

2022-09-25T15:34:15Z

Vipul:

This privacy policy is common to subject wikis. For the original privacy policy, refer [[Ref:Ref:Privacy policy]].

==Privacy for readers==

If you are surfing this website, your actions are logged in our usage logs. These usage logs are accessible to:

* The site's administrators and technical support group. For a full list of administrators, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com.
* The service that hosts the data and servers, which is currently [http://www.linode.com Linode].
* Google Analytics, which has been integrated to collect site statistics. View Google's privacy policy here: http://www.google.com/intl/en_ALL/privacypolicy.html
* Other third-party JS scripts that collect user activity; none of these should collect any personally identifiable information (PII). For a list of all scripts running at the current time, contact [[User:Vipul|Vipul Naik]] by email: vipulnaik1@gmail.com.

==Privacy for editors==

Editing on subject wikis is generally permitted only for registered users. Registered users must, at the time of registration, provide their real name, and enter basic information about their reason for interest. ''No'' private information such as date of birth, social security or taxation number, or home address is sought.

Regarding personal information:

* The email IDs of registered users are visible to site administrators only. For information about site administrators, contact vipulnaik1@gmail.com with the particular subject wiki and the reason for request.
* All editing activity by registered users is recorded on the site and is visible to all site users. However, this information is not indexed by search engines that follow robots.txt.
* For edits made by registered users when logged in, the originating IP addresses for the edits can be accessed only by the site administrators.
* Passwords chosen by registered users are not humanly accessible, even to site administrators.

User:Vipul/Sandbox

2017-11-27T07:21:53Z

Vipul: Created page with "<math>e^{\pi^2 + \sqrt{2}}</math>"

User:Vipul/sandbox

2016-09-05T20:55:54Z

Vipul: Created page with "<math>\frac{e^{\pi}}{\sqrt{t^2 + 1}}</math>"

Template:Top notice

2015-04-17T23:55:27Z

Vipul:

{{quotation|Welcome to '''{{fullsitetitle}}'''. This is a pre-pre-alpha stage commutative algebra wiki primarily managed by [[User:Vipul|Vipul Naik]], a Ph.D. in Mathematics at the University of Chicago. It is part of a broader subject wikis initiative -- see the [[Ref:Main Page|subject wikis reference guide]] for more details.}}

Ring of Gaussian integers

2014-01-29T21:07:31Z

Vipul: Created page with "{{particular curing}} ==Definition== The '''ring of Gaussian integers''' <math>\mathbb{Z}[i]</math> is defined in the following ways: # It is the subring generated by the [..."

{{particular curing}}

==Definition==

The '''ring of Gaussian integers''' <math>\mathbb{Z}[i]</math> is defined in the following ways:

# It is the subring generated by the [[ring of rational integers]] and the element <math>i</math> (a square root of -1) in the [[field of complex numbers]].
# It is the integral extension <math>\mathbb{Z}[t]/(t^2 + 1)</math> of the [[ring of rational integers]] <math>\mathbb{Z}</math>, with the image of the indeterminate <math>t</math> denoted as <math>i</math>.
# It is the [[ring of integers in a number field|ring of integers]] in the [[number field]] <math>\mathbb{Q}(i)</math>, a quadratic extension of the rationals given as <math>\mathbb{Q}[t]/(t^2 + 1)</math> (with the image of <math>t</math> denoted <math>i</math>).

==Ring properties==

{| class="sortable" border="1"
! Property !! Meaning !! Satisfied? !! Explanation
|-
| [[satisfies property::integral domain]] || product of nonzero elements is nonzero || Yes || Follows from being a subring of <math>\mathbb{C}</math>
|-
| [[satisfies property::Euclidean domain]] || has a [[Euclidean norm]] || Yes || In fact, the standard algebraic norm (which in this case is the same as the square of the complex modulus) is a Euclidean norm. The key geometric fact used is that the distance of any point in <math>\mathbb{C}</math> from the closest point in <math>\mathbb{Z}[i]</math> is less than 1.
|-
| [[satisfies property::principal ideal domain]] || [[integral domain]] and every [[ideal]] in it is a [[principal ideal]] || Yes || See [[Euclidean implies PID]]
|-
| [[satisfies property::unique factorization domain]] || every element has a unique factorization into irreducibles up to units || Yes || See [[PID implies UFD]]
|-
| [[satisfies property::Noetherian domain]] ||[[integral domain]] || Yes || Follows from being a PID
|-
| [[satisfies property::Bezout domain]] || || Yes || Follows from being a PID
|-
| [[satisfies property::Dedekind domain]] || || Yes || Follows from being a PID
|}

Template:Particular curing

2014-01-29T20:55:46Z

Vipul: Created page with "{{quotation|This article defines a particular commutative unital ring.<br>See all particular commutative unital rings}}<i..."

{{quotation|This article defines a particular [[commutative unital ring]].<br>[[:Category:Particular commutative unital rings|See all particular commutative unital rings]]}}<includeonly>[[Category:Particular commutative unital rings]]</includeonly>

Ring of rational integers

2014-01-29T20:54:40Z

Vipul:

{{particular curing}}

==Definition==

The ring <math>\mathbb{Z}</math>, called the '''ring of rational integers''' or sometimes simply the '''ring of integers''', is the ring whose elements are the rational integers, with the usual addition and multiplication. Explicitly, the underlying set is <math>\{ \dots, -2, -1, 0, 1, 2, \dots \}</math> and the addition and multiplication are the usual ones.

The adjective ''rational'' is used in the name in circumstances where there may be potential confusion with the [[ring of integers in a number field]].

This ring is the initial object in the category of [[commutative unital ring]]s.

==Ring properties==

{| class="sortable" border="1"
! Property !! Meaning !! Satisfied? !! Explanation
|-
| [[satisfies property::integral domain]] || zero is not a product of nonzero elements || Yes ||
|-
| [[satisfies property::Euclidean domain]] || admits a [[Euclidean norm]] || Yes || See [[ring of rational integers is Euclidean with norm equal to absolute value]] (the standard choice of norm is the absolute value); see also [[ring of rational integers is Euclidean with norm equal to binary logarithm of absolute value]]
|-
| [[satisfies property::principal ideal domain]] (PID) || every [[ideal]] is a [[principal ideal]] || Yes || Follows from being Euclidean and [[Euclidean implies PID]]
|-
| [[satisfies property::unique factorization domain]] || every element has a unique factorization into irreducibles (same as primes) up to units || Yes || Follows from being a PID and [[PID implies UFD]]
|-
| [[satisfies property::Noetherian domain]] ||[[integral domain]] || Yes || Follows from being a PID
|-
| [[satisfies property::Bezout domain]] || || Yes || Follows from being a PID
|-
| [[satisfies property::Dedekind domain]] || || Yes || Follows from being a PID
|-
| [[satisfies property::interpolation domain]] || || Yes || For any <math>n</math>, there exists a tuple of <math>n + 1</math> elements such that evaluation at these defines a bijection between the polynomials of degree at most <math>n</math> in the [[ring of integer-valued polynomials]] and <math>R^{n+1}</math>.
|}

Ring of integers

2014-01-29T20:38:18Z

Vipul:

You might be looking for:

* [[Ring of rational integers]] <math>\mathbb{Z}</math>, the initial object in the category of [[commutative unital ring]]s.
* [[Ring of integers in a number field]], defined in the context of a [[number field]].

Ring of rational integers

2014-01-29T20:37:24Z

Vipul: Created page with "==Definition== The ring <math>\mathbb{Z}</math>, called the '''ring of rational integers''' or sometimes simply the '''ring of integers''', is the ring whose elements are the..."

==Definition==

The ring <math>\mathbb{Z}</math>, called the '''ring of rational integers''' or sometimes simply the '''ring of integers''', is the ring whose elements are the rational integers, with the usual addition and multiplication. Explicitly, the underlying set is <math>\{ \dots, -2, -1, 0, 1, 2, \dots \}</math> and the addition and multiplication are the usual ones.

The adjective ''rational'' is used in the name in circumstances where there may be potential confusion with the [[ring of integers in a number field]].

Z

2014-01-29T20:35:37Z

Vipul: Redirected page to Ring of rational integers

#redirect [[Ring of rational integers]]

Euclidean domain

2014-01-29T20:34:00Z

Vipul: /* Relation with other properties */

{{integral domain property}}

==Definition==

===Symbol-free definition===

An [[integral domain]] is said to be '''Euclidean''' if it admits a [[Euclidean norm]].

===Definition with symbols===

An [[integral domain]] <math>R</math> is termed a '''Euclidean domain''' if there exists a function <math>N</math> from the set of nonzero elements of <math>R</math> to the set of nonnegative integers satisfying the following properties:

* <math>N(x) = 0</math> if and only if <math>x</math> is a unit
* Given nonzero <math>a</math> and <math>b</math> in <math>R</math>, there exist <math>q</math> and <math>r</math> such that <math>a = qb + r</math> and either <math>r = 0</math> or <math>N(r) < N(b)</math>.

We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.

Such a function <math>N</math> is called a [[Euclidean norm]] on <math>R</math>.

===Caveats===

* The definition of Euclidean norm does not require the ring to be an integral domain. A commutative unital ring that admits a Euclidean norm is termed a Euclidean ring.
* The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].

==Examples==

===Standard examples===

* The [[ring of rational integers]] <math>\mathbb{Z}</math> is a Euclidean domain with Euclidean norm defined by the absolute value. {{proofat|[[Ring of integers is Euclidean with norm equal to absolute value]]}}
* The [[polynomial ring over a field]] <math>k[x]</math> is a Euclidean domain with Euclidean norm defined by the degree of a polynomial. This is, in fact a ''uniquely'' Euclidean norm. and hence the polynomial ring over a field is a uniquely Euclidean domain. {{proofat|[[Polynomial ring over a field is uniquely Euclidean with norm equal to degree]]}}

===Other examples===

* The [[ring of Gaussian integers]] <math>\mathbb{Z}[i]</math> is a Euclidean domain with Euclidean norm equal to the norm in the sense of a quadratic integer ring. {{proofat|[[Ring of Gaussian integers is norm-Euclidean]]}}
* A quadratic integer ring, or more generally, a [[ring of integers in a number field]], is termed [[norm-Euclidean ring of integers in a number field]] if it is Euclidean with respect to the algebraic norm. Since there is a correspondence between number fields and their rings of integers, we often abuse language and say that the number field itself is norm-Euclidean.
* Any [[discrete valuation ring]] is a Euclidean domain where the norm of an element is given by the largest power of the irreducible that divides it. For instance, the [[formal power series ring over a field]] is a Euclidean domain, where the norm of a formal power series is the smallest <math>n</math> for which the coefficient of <math>x^n</math> that is nonzero.

===Pathological examples===

On a field, ''any'' norm function is Euclidean. This is because we can always choose a quotient so that the remainder is zero.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::uniquely Euclidean domain]] || there is a [[Euclidean norm]] for which Euclidean division is unique. || || || {{intermediate notions short|Euclidean domain|uniquely Euclidean domain}}
|-
| [[Weaker than::Polynomial ring over a field]] || it can be written as the [[polynomial ring]] <math>K[x]</math> for a [[field]] <math>K</math>. || || || {{intermediate notions short|Euclidean domain|polynomial ring over a field}}
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::multi-stage Euclidean domain]] || || || || {{intermediate notions short|multi-stage Euclidean domain|Euclidean domain}}
|-
| [[Stronger than::principal ideal domain]] || [[integral domain]] that is a [[principal ideal ring]] || [[Euclidean implies PID]] || [[PID not implies Euclidean]] || {{intermediate notions short|principal ideal domain|Euclidean domain}}
|-
| [[Stronger than::Bezout domain]] || [[integral domain]] in which every [[finitely generated ideal]] is [[principal ideal|principal]] || || || {{intermediate notions short|Bezout domain|Euclidean domain}}
|-
| [[Stronger than::unique factorization domain]] || || || || {{intermediate notions short|unique factorization domain|Euclidean domain}}
|-
| [[Stronger than::Dedekind domain]] || Noetherian, normal, one-dimensional domain || || || {{intermediate notions short|Dedekind domain|Euclidean domain}}
|-
| [[Stronger than::Noetherian domain]] || [[integral domain]] and every ideal is finitely generated || || || {{intermediate notions short|Noetherian domain|Euclidean domain}}
|-
| [[Stronger than::Noetherian ring]] || every ideal is finitely generated || || || {{intermediate notions short|Noetherian ring|Euclidean domain}}
|}

===Properties of Euclidean norms===

Euclidean norms can in general be very weirdly behaved, but some Euclidean norms are good. For a complete list of properties of Euclidean norms (i.e., properties against which a given Euclidean norm can be tested), refer:

[[:Category:Properties of Euclidean norms]]

Here are some important properties that most ''typical'' Euclidean norms satisfy:

* [[Multiplicatively monotone Euclidean norm]]

==Metaproperties==

{{not poly-closed curing property}}

The polynomial ring over a Euclidean domain need not be a Euclidean domain. One example is the polynomial ring with integer coefficients, which is not a Euclidean domain; another example is the polynomial ring in ''two'' variables over a field (which can be viewed as the polynomial ring in one variable, over the polynomial ring over a field).