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* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

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## Some Related Sentences

spectrum and any

An artist

**is**a person engaged**in**one or more**of**__any__**of**a broad__spectrum__**of**activities related to creating art**,**practicing**the**arts and / or demonstrating an art.
Black

**is****the**color**of**objects**that**do not emit or reflect light**in**__any__part**of****the**visible__spectrum__; they absorb**all**such frequencies**of**light.**The**

__spectrum__

**of**

__any__element x

**is**a closed subset

**of**

**the**closed disc

**in**C

**with**radius || x || and center 0

**,**and thus

**is**

**compact**.

*****

**The**

__spectrum__

**of**

__any__bounded linear operator on a Banach space

**is**a nonempty

**compact**subset

**of**

**the**complex numbers C. Conversely

**,**

__any__

**compact**subset

**of**C arises

**in**this manner

**,**as

**the**

__spectrum__

**of**some bounded linear operator.

For instance

**,**a diagonal operator on**the**Hilbert space may have__any__**compact**nonempty subset**of**C as__spectrum__.
Spectroscopic studies revealed absorption lines

**in****the**Jovian__spectrum__due to diatomic sulfur**(**S < sub > 2 </ sub >) and carbon disulfide**(**CS < sub > 2 </ sub >),**the**first detection**of**either**in**Jupiter**,**and only**the**second detection**of**S < sub > 2 </ sub >**in**__any__astronomical object.
These are capable

**of****the**most severe types**of**molecular damage**,**which can happen**in**biology to__any__type**of**biomolecule**,**including mutation and cancer**,**and often at great depths from**the**skin**,**since**the**higher end**of****the**X-ray__spectrum__**,**and**all****of****the**gamma ray__spectrum__**,**are penetrating to matter.
Fascists have commonly opposed having a firm association

**with**__any__section**of****the**left-right__spectrum__**,**considering it inadequate to describe their beliefs**,**though fascism's goal to promote**the**rule**of**people deemed innately superior while seeking to purge society**of**people deemed innately inferior**is**identified as a prominent far-right theme.
Individuals were categorized according to their so-called " rejection

__spectrum__" which allowed doctors to counter__any__immune system responses to**the**new organs**,**allowing transplants to " take " for life.
We can also show

**that****the**possible values**of****the**observable A**in**__any__state must belong to**the**__spectrum__**of**A.
In linguistics

**the**term orthography**is**often used to refer to__any__method**of**writing a language**,**without judgment as to right and wrong**,****with**a scientific understanding**that**orthographic standardization exists on a__spectrum__**of**strength**of**convention.
Pacifism covers a

__spectrum__**of**views ranging from**the**belief**that**international disputes can and should be peacefully resolved ; to calls for**the**abolition**of****the**institutions**of****the**military and war ; to opposition to__any__organization**of**society through governmental force**(**anarchist or libertarian pacifism ); to rejection**of****the**use**of**physical violence to obtain political**,**economic or social goals ; to opposition to violence under__any__circumstance**,**including defense**of**self and others.
Pacifism covers a

__spectrum__**of**views**,**including**the**belief**that**international disputes can and should be peacefully resolved**,**calls for**the**abolition**of****the**institutions**of****the**military and war**,**opposition to__any__organization**of**society through governmental force**(**anarchist or libertarian pacifism ), rejection**of****the**use**of**physical violence to obtain political**,**economic or social goals**,****the**obliteration**of**force**except****in****cases**where it**is**absolutely necessary to advance**the**cause**of**peace**,**and opposition to violence under__any__circumstance**,**even defense**of**self and others.
If

**the**surface has__any__transparent or translucent properties**,**it refracts a portion**of****the**light beam into itself**in**a different direction while absorbing some**(**or**all****)****of****the**__spectrum__**(**and possibly altering**the**color**).**
Fundamental strings exist

**in**9 dimensions and**the**strings can vibrate**in**__any__direction**,**meaning**that****the**__spectrum__**of**vibrational modes**is**much richer.
As

**with**__any__religious movement**,**a theological__spectrum__exists within Adventism comparable to**the**fundamentalist-moderate-liberal__spectrum__**in****the**wider Christian church and**in**other religions.
In general

**,**__any__particular instrument will operate over a small portion**of**this total range because**of****the**different techniques used to measure different portions**of****the**__spectrum__.

spectrum and commutative

In abstract algebra and algebraic geometry

**,****the**__spectrum__**of**a__commutative__**ring**R**,**denoted by Spec**(**R ),**is****the****set****of****all**proper**prime****ideals****of**R. It**is**commonly augmented**with****the****Zariski****topology**and**with**a structure sheaf**,**turning it into a locally ringed space.
As a rule

**of**thumb**,****any**sort**of**construction**that**takes as input a fairly general object**(**often**of**an algebraic**,**or topological-algebraic nature**)**and outputs a**compact**space**is**likely to use Tychonoff: e. g.,**the**Gelfand space**of**maximal**ideals****of**a__commutative__C*****algebra**,****the**Stone space**of**maximal**ideals****of**a Boolean algebra**,**and**the**Berkovich__spectrum__**of**a__commutative__Banach**ring**.
If X

**is**an algebraic variety carrying**the****Zariski****topology****,**we can define a locally ringed space by taking O < sub > X </ sub >( U**)**to be**the****ring****of**rational functions defined on**the**Zariski-open**set**U which do not blow up**(**become infinite**)**within U.**The**important generalization**of**this example**is****that****of****the**__spectrum__**of****any**__commutative__**ring**; these spectra are also locally ringed spaces.
Let P be a finitely generated projective module over a

__commutative__**ring**R and X be**the**__spectrum__**of**R.**The**rank**of**P at a**prime**ideal**in**X**is****the**rank**of****the**free-module.
He defined

**the**__spectrum__**of**a__commutative__**ring**as**the**space**of****prime****ideals****with****Zariski****topology****,****but**augments it**with**a sheaf**of**rings: to every Zariski-open**set**he assigns a__commutative__**ring****,**thought**of**as**the****ring****of**" polynomial functions " defined on**that****set**.**The**functor associates to every

__commutative__

**ring**its

__spectrum__

**,**

**the**scheme defined by

**the**

**prime**

**ideals**

**of**

**the**

**ring**.

Zero divisors have a topological interpretation

**,**at least**in****the**case**of**__commutative__rings: a**ring**R**is**an integral domain**,**if and only if it**is**reduced and its__spectrum__Spec R**is**an irreducible topological space.
Any affine group scheme

**is****the**__spectrum__**of**a__commutative__Hopf algebra**(**over a base S**,**this**is**given by**the**relative__spectrum__**of**an O < sub > S </ sub >- algebra**).**
For an arbitrary group scheme G

**,****the****ring****of**global sections also has a__commutative__Hopf algebra structure**,**and by taking its__spectrum__**,**one obtains**the**maximal affine quotient group.
In mathematics

**,**a spectral space**is**a topological space which**is**homeomorphic to**the**__spectrum__**of**a__commutative__**ring**.**The**

**prime**

__spectrum__Spec

**(**R

**)**

**of**a

__commutative__

**ring**R

**with**

**the**

**Zariski**

**topology**

**is**a

**compact**sober T < sub > 0 </ sub > space.

For example

**,**there**is**a duality between__commutative__rings and affine schemes: to every__commutative__**ring**A there**is**an affine__spectrum__**,**Spec A**,**conversely**,**given an affine scheme S**,**one gets back a**ring**by taking global sections**of****the**structure sheaf O < sub > S </ sub >.
In functional analysis

**,**a branch**of**mathematics**,****the**Borel functional calculus**is**a functional calculus**(****that****is****,**an assignment**of**operators from__commutative__algebras to functions defined on their__spectrum__), which has particularly broad scope.*****In

__commutative__algebra

**,**a

__commutative__

**ring**R

**is**irreducible if its

**prime**

__spectrum__

**,**

**that**

**is**

**,**

**the**topological space Spec R

**,**

**is**an irreducible topological space.

A complex orientation on an associative

__commutative__**ring**__spectrum__E**is**an element x**in**E < sup > 2 </ sup >( CP < sup >∞</ sup >) whose restriction to E < sup > 2 </ sup >( CP < sup > 1 </ sup >)

spectrum and ring

*****

**the**

**Zariski**

**topology**on an algebraic variety or on

**the**

__spectrum__

**of**a

__ring__

**,**used

**in**algebraic geometry

**,**

It follows readily from

**the**definition**of****the**__spectrum__**of**a__ring__**,****the**space**of****prime****ideals****of**equipped**with****the****Zariski****topology****,****that****the**Krull dimension**of****is**precisely equal to**the**irreducible dimension**of**its__spectrum__.*****In

**the**category

**of**schemes

**,**Spec

**(**Z

**)**

**the**

**prime**

__spectrum__

**of**

**the**

__ring__

**of**integers

**is**a terminal object.

Modern algebraic geometry takes

**the**__spectrum__**of**a__ring__**(****the****set****of**proper**prime****ideals****)**as its starting point.
Thus

**,**V**(**S**)****is**"**the**same as "**the**maximal**ideals**containing S. Grothendieck's innovation**in**defining Spec was to replace maximal**ideals****with****all****prime****ideals**;**in**this formulation it**is**natural to simply generalize this observation to**the**definition**of**a closed**set****in****the**__spectrum__**of**a__ring__.*****Spec k

**,**

**the**

__spectrum__

**of**

**the**polynomial

__ring__over a field k

**,**which

**is**also denoted

**,**

**the**affine line:

**the**polynomial

__ring__

**is**known to be a principal ideal domain and

**the**irreducible polynomials are

**the**

**prime**elements

**of**k. If k

**is**algebraically closed

**,**for example

**the**field

**of**complex numbers

**,**a non-constant polynomial

**is**irreducible if and only if it

**is**linear

**,**

**of**

**the**form t − a

**,**for some element a

**of**k. So

**,**

**the**

__spectrum__consists

**of**one closed point for every element a

**of**k and a generic point

**,**corresponding to

**the**zero ideal.

Just as

**in**classical algebraic geometry**,****any**__spectrum__or projective__spectrum__**is****compact****,**and if**the**__ring__**in**question**is**Noetherian then**the**space**is**a Noetherian space.
In

**the**NMR__spectrum__**of**a dimethyl derivative**,**two nonequivalent signals are found for**the**two methyl groups indicating**that****the**molecular conformation**of**this cation not perpendicular**(**as**in**A**)****but****is**bisected**(**as**in**B**)****with****the**empty p-orbital and**the**cyclopropyl__ring__system**in****the**same plane:
A scheme

**is**a locally ringed space such**that**every point has a neighbourhood**,**which**,**as a locally ringed space**,****is**isomorphic to a__spectrum__**of**a__ring__.
If

**the**same signal**is**sent to both inputs**of**a__ring__modulator**,****the**resultant harmonic__spectrum__**is****the**original frequency domain doubled**(**if f < sub > 1 </ sub >
An equivalent

**but**streamlined construction**is**given by**the**Proj construction**,**which**is**an analog**of****the**__spectrum__**of**a__ring__**,**denoted " Spec ", which defines an affine scheme.0.234 seconds.